Set Theory and Logic

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Recent submissions

Any replacements are listed farther down

[586] viXra:1912.0125 [pdf] submitted on 2019-12-06 13:56:15

Refutation of Perfectly Parallel Fairness Certification of Neural Networks

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

e evaluate the seminal definition of causal fairness which is not tautologous. This refutes the conjecture of perfectly parallel fairness certification of neural networks. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[585] viXra:1912.0111 [pdf] submitted on 2019-12-06 06:03:47

Refutation of Duty to Warn in Strategic Games

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate two axioms for duty to warn modalities: negative introspection from epistemic logic S5 and the conjecture introspection of blameworthiness. Both are not tautologous and refutations. This forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[584] viXra:1912.0091 [pdf] submitted on 2019-12-04 21:38:28

Refutation of Disjunction as Defined in Alternative Herbrand Functional Interpretation

Authors: Colin James III
Comments: 1 Page.

For alternative formulation of the Herbrand functional interpretation, we evaluate disjunction as defined from the other connectives and quantifiers which is not tautologous. This refutes the conjecture to simplify types of witnesses in the Herbrand functional interpretation, avoiding the use of “sets of functionals”. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[583] viXra:1912.0075 [pdf] submitted on 2019-12-04 04:59:47

Refutation of Flow Theory

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate two theorems, one postulate, and nine examples of ZFC axioms as not tautologous. This refutes flow theory. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[582] viXra:1912.0026 [pdf] submitted on 2019-12-02 14:28:34

Refutation of Abstract Argumentation and the Rational Man

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the rational man in abstract argumentation. It is not tautologous, hence refuting its bounded rationality in choice. This forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[581] viXra:1912.0022 [pdf] submitted on 2019-12-01 21:06:56

Refutation of Axiomatisability of the Class of Monolithic Groups in Nilpotent Groups

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate main theorem of the paper which is not tautologous, refuting the conjecture of axiomatisability of the class of monolithic groups in a variety of nilpotent groups. This forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[580] viXra:1911.0488 [pdf] submitted on 2019-11-29 06:32:40

Logic Cila (Da Costa’s C1) Confirmed as Algebraizable, Refuting Paraconsistent Set Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

For logic Cila, we evaluate the implication connective truth table from 2001 which is not tautologous. This means the implication connective is algebraizable and refuted, and further refutes paraconsistent logic, and the derived conjecture of paraconsistent set theory as based thereon from 2019. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[579] viXra:1911.0464 [pdf] submitted on 2019-11-27 13:35:40

Refutation of Leibniz’ Indiscernibility of Identicals and Identity of Indiscernibles as Equivalent

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate Leibniz’ law(s) of indiscernibility of identicals and identity of indiscernibles which are not equivalents, and the latter is not tautologous. This contradicts Leibniz’ law of identity of indiscernibles. We also note that second-order expressions are expressible as first-order expressions because of the equivalence of respective quantifiers and modal operators in VŁ4. The results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[578] viXra:1911.0461 [pdf] submitted on 2019-11-26 19:28:46

Refutation of More Axioms: Tarski’s; ax-Groth; Grothpw; Limitation of Size; and Class of All Sets

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate Tarski’s axiom, ax-groth, grothpw, the axiom of the limitation of size, and the class of all sets as not tautologous. This further refutes the Tarski–Grothendieck set theory. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[577] viXra:1911.0434 [pdf] submitted on 2019-11-25 10:11:25

Refutation of Recursive Enumerability and Elementary Frame Definability in Predicate Modal Logic

Authors: Colin James III
Comments: 4 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate two equations as not tautologous to show the standard translation of a predicate model into a logic language fails. We evaluate five equations as not tautologous, with the last result as contradictory, to show a class defined to contain frames fails to satisfy a classical first-order formula or to claim logic system equivalency. The conjecture of recursive enumerability and elementary frame definability in predicate modal logic is refuted. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[576] viXra:1911.0416 [pdf] submitted on 2019-11-24 17:03:44

Confirmation of Modus Inversus as a bi-Valent Connective (And Denial of Its Refutation)

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate modus inversus, confirming it as a bi-valent connective. This denies its anonymous refutation to form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[575] viXra:1911.0412 [pdf] submitted on 2019-11-23 22:35:34

Recent Advances in Meth8/VŁ4, a Modal Model Checker with Universal Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Updated abstract at ersatz-systems.com.

We use the Meth8/VŁ4 modal logic model checker to evaluate 696 artifacts in 3607 assertions to refute 3053 as non tautologous (84.6%). This is the two page abstract only. The current document in 1156-pages includes the availability of Meth8/VŁ4 and is requestable from info@ersatz-systems.com.
Category: Set Theory and Logic

[574] viXra:1911.0411 [pdf] submitted on 2019-11-23 22:57:47

Refutation of Brouwer’s Dense Linear Orderings (Dlo)

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate three axioms of Brouwer’s dense linear orderings (DLO) as not tautologous, hence that as a basis for refuting equality and equivalence, intuitionistically. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[573] viXra:1911.0390 [pdf] submitted on 2019-11-23 06:58:00

Number Theory Beyond Frege on the Necessity of Open Arity

Authors: Hannes Hutzelmeyer
Comments: 13 Pages.

A closer look at mathematical proofs led Gottlob Frege to realize that Aristotle's syllogism logic was not sufficient for many theorems. He developped what today is called first-order predicate logic. It is usually thought that predicate logic is sufficient for the theory of natural numbers. However, this first step of modern logic development again is not sufficient. One needs another step, especially to allow for so-called open arity of arrays. This second step cannot be done in general in object-language based on predicate logic but only by metalanguage. Therefore one needs something like the FUME-method (put forward by the author) which allows for a precise treatment of both language levels. Dot-dot-dot … is not admissible in predicate logics as it needs some kind of recursion. In metalanguage, however, one has to introduce some basic recursion right from the setup (but it is much weaker than primitive recursion). For natural numbers two examples are given, one for a concrete version of Robinson arithmetic and one for recursive arithmetic. Without the second step to metalanguage one cannot express some of the most important so-called theorems of number theory in a direct fashion, leave alone prove them. Actually some are not theorems but metatheorems. The examples comprise Chinese remainders, Gödel's beta-function, little Gauss's summing up of numbers, Euclid's unlimited primes and the canonical representation of a natural number (fundamental theorem of natural arithmetic). After one has included the second step which allows one to talk about open arities in metalanguage one can tackle the problem of talking about number-arrays in object language. One can do this to a certain extent by coding number-arrays by (usually) two numbers. This can be done even in Robinson arithmetic using 'Gödel's beta-function'. But one has to make use of the second step before one can return to object-language. Of course, the introduction of two tiers, i.e. object-language and metalanguage, is necessary for many other areas of mathematics, if not to say, most of them.
Category: Set Theory and Logic

[572] viXra:1911.0375 [pdf] submitted on 2019-11-22 02:36:08

Непригодность общепринятых стереотипных тавтологий классической математики для доказательства теорем при классическом подходе к доказательству теорем

Authors: Viktoria Kondratenko
Comments: 9 Pages.

Оторванность гипотетических теорий от реалий живой материи стала причиной проникновения мистики в научные теории. При мистическом мышлении идея применения аналитического метода решения задач познания в голову не приходит. Диалектическая логика в отличие от мистики утверждает обратное: любые проблемные задачи познания жизнедеятельности процессов и явлений мироздания разрешимы исключительно аналитическим путѐм, при этом единственным методом. Автором создана универсальная и формальная теория решения интеллектуальных (т.е. не имеющих заранее известных алгоритмов решения) задач, связанных с познанием жизнедеятельности естественных и рукотворных процессов в любых явлениях мироздания – метод аксиоматического моделирования Кондратенко, эффективность которого достигается путѐм корректной постановки задачи и еѐ решения чисто формальным методом. Корректность постановки задачи означает, прежде всего, признание несостоятельности всех гипотетических (не подтверждѐнных результатами натурного экспериментирования с предметом познания) теорий. В качестве примера в статье рассматривается парадокс при классическом подходе к доказательству теорем, состоящий в непригодности общепринятых стереотипных тавтологий классической математики для доказательства теорем.
Category: Set Theory and Logic

[571] viXra:1911.0374 [pdf] submitted on 2019-11-22 02:39:06

Формальная модель диалектической логики управления жизнеобеспечивающими функциями психики человека

Authors: Viktoria Kondratenko
Comments: 9 Pages.

Диалектическая логика управления любыми функциональными системами организма человека вытекает из концептуальных знаний о жизнедеятельности этих функциональных систем, добытых исключительно путём натурного экспериментирования с ними. Поэтому концептуальные знания о жизнедеятельности каждой исследуемой функциональной системы человека должны предшествовать описанию сущности логики управления ею. Психика человека является одной из важнейших функциональных систем организма человека. В статье представлены основополагающие концепты жизнедеятельности этой системы, которые позволяют выявить сущность диалектической логики управления её жизнеобеспечивающими функциями, и формальная модель диалектической логики управления функциональной системой психики человека
Category: Set Theory and Logic

[570] viXra:1911.0350 [pdf] submitted on 2019-11-20 09:59:46

Refutation of Premise Rule in Intuitionistic Set Theories

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the seminal equation of the conjecture which is not tautologous, hence refuting the premise rule in intuitionistic set theories. This forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[569] viXra:1911.0334 [pdf] submitted on 2019-11-18 23:21:45

Refutation of Justification Based Reasoning for Dynamic Conflict Resolution

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We first evaluate the last of seven examples, which is not tautologous. This refutes the conjecture of justification based reasoning for dynamic conflict resolution, to form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[568] viXra:1911.0320 [pdf] submitted on 2019-11-18 08:50:16

A Refutation Of Ilija Barukˇci ́c’s Modus Inversus AndConclusions Following From It

Authors: Nhef Luminati
Comments: 6 Pages.

In this paper the general validity of the methodological approach of frequent vixra.org contributor Ilija Barukčić are examined and refuted.
Category: Set Theory and Logic

[567] viXra:1911.0244 [pdf] submitted on 2019-11-13 23:51:00

Refutation of Bunch Theory with Derivations

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate bunch theory in ten equations which are not tautologous. This refutes the conjecture of bunch theory along with its derivations of: improper bunch theory; wholistic functions; fixed point theory; and formal grammars. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[566] viXra:1911.0215 [pdf] submitted on 2019-11-12 05:59:15

Refutation of Reduction of Provability Logics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate definitions for “Lob” and the “completeness principle” also the necessitation rule. None is tautologous. Hence reduction of provability logics is not confirmed. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[565] viXra:1911.0204 [pdf] submitted on 2019-11-11 20:09:57

Refutation of H-Inductive Sentences and Positive Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate h-universal sentences which are not tautologous. This refutes h-inductive sentences and positive logic. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[564] viXra:1911.0183 [pdf] submitted on 2019-11-10 10:39:24

Refutation of Tense Logic and Display to Labelled Proofs Back Again for Tense Logics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate tense logic (Kt) as an extension of K. The equation for A→□A is not tautologous, hence refuting Kt. What follows is that the conjecture of display to labelled proofs and back again for tense logic is not confirmed. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[563] viXra:1911.0068 [pdf] submitted on 2019-11-04 16:22:34

Refutation of Multi-Valued Concurrent Dynamic Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate two operators for parallel and sequential composition which are not tautologous. The conjecture of multi-valued, concurrent, propositional, dynamic logics fails. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[562] viXra:1911.0030 [pdf] submitted on 2019-11-02 07:11:49

Refutation of Cohen’s First Model and Derivations as the Method of Forcing

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the definition of Cohen’s first model as not tautologous, hence refuting it and impugning derived conjectures such as forcing. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[561] viXra:1911.0022 [pdf] submitted on 2019-11-01 13:19:50

Refutation of a Common Expression for Law of Trichotomy, Replaced by Shortest Known Proof

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We show a common formal logic expression for the law of trichotomy is not tautologous, and replace it with the shortest known proof. The first results forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[560] viXra:1911.0021 [pdf] submitted on 2019-11-01 13:46:34

Refutaton of Frege’s Inference Rule of Generalization

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate Frege’s inference rule of generalization which is not tautologous. This result forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[559] viXra:1911.0015 [pdf] submitted on 2019-11-01 02:10:03

Refutation of Differential Hoare Logics with Isabelle/hol

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

e evaluate the ordering of K essential to Kleene algebra with test (KAT). It is not tautologous, but the basis for other conjectures: differential dynamic logic (dL); differential Hoare logic; Morgan-style differential refinement calculus; “reasoning with evolution”; and verification conjectures formalized in Isabelle/HOL. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[558] viXra:1910.0619 [pdf] submitted on 2019-10-29 14:23:22

Refutation of Lifting Countable to Uncountable Arithmetic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

In reverse mathematics (RM) we evaluate an axiom which is not tautologous. Because RM derives theorems from axioms, if an axiom is refuted, then its derived theorem is also refuted, hence refuting RM itself and derived conjectures such as “uplifting” countable mathematics to uncountable mathematics. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[557] viXra:1910.0618 [pdf] submitted on 2019-10-29 14:25:19

Refutation of the Law of Excluded Middle as a no-go Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the seminal theorem to reverse mathematics (RM). It is not tautologous, hence refuting RM. By extension the following are also not confirmed: the Tietze extension theorem; Ekeland’s variational principle; and the law of excluded middle as a no-go theorem. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[556] viXra:1910.0616 [pdf] submitted on 2019-10-29 19:05:04

Refutation of the Two Couriers Problem

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We rewrite De Morgan’s framing of the two courier’s problem to avoid his six cases and in one sentence which is not tautologous. The conjectured problem is not a theorem (or contradiction), but not well formed either, and hence unsolvable as stated. The value of zero is not directly tested and not in a fractional expression, so division by zero is not relevant. These results are a non tautologous fragment of logic VŁ4.
Category: Set Theory and Logic

[555] viXra:1910.0614 [pdf] submitted on 2019-10-29 22:24:58

Refutation of Computerized Categorical Axiomatization of the Modeloid

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the Wagner-Preston representation theorem for injective homomorphism to map the modeloidal concept, as an inverse semigroup, from a categorical perspective, as envisioned by D. S. Scott. The theorem is not tautologous, refuting also the Isabelle/HOL tool of C. Benzmüller used to prove it. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[554] viXra:1910.0595 [pdf] submitted on 2019-10-28 15:17:33

Refutation of Deep Reinforcement Learning in Hol4

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate rewriting terms as based on axioms in a subset of Robinson arithmetic on which deep reinforcement learning in HOL4 relies: two such axioms as not tautologous. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[553] viXra:1910.0559 [pdf] submitted on 2019-10-27 07:25:40

Instruction Set Completeness Theorem: Concept, Relevance, Proof, and Example for Processor Architecture

Authors: William F. Gilreath
Comments: 22 Pages. Published in the General Science Journal

The Instruction Set Completeness Theorem is first formally defined and discussed in the seminal work on one-instruction set computing—the book Computer Architecture: A Minimalist Perspective. Yet the original formalism of the Instruction Set Completeness Theorem did not provide a definitive, explicit mathematical proof of completeness, analyze both singular and plural instruction sets that were either complete or incomplete, nor examine the significance of the theorem to computer architecture instruction sets. A mathematical proof of correctness shows the equivalence of the Instruction Set Completeness Theorem to a Turing machine, a hypothetical model of computation, and thereby establishes the mathematical truth of the Instruction Set Completeness Theorem. With a more detailed examination of the Instruction Set Completeness Theorem develops several surprising conclusions for both the instruction set completeness theorem, and the instruction sets for a computer architecture.
Category: Set Theory and Logic

[552] viXra:1910.0556 [pdf] submitted on 2019-10-27 08:00:56

Modus Inversus – if (Premise is False) Then (Conclusion is False)

Authors: Ilija Barukčić
Comments: 9 pages. Copyright © 2018 by Ilija Barukčić, Jever, Germany. Published by:

Objective: When theorems or theories are falsified by a formal prove or by observations et cetera, authors respond many times by different and sometimes inappropriate counter-measures. Even if the pressure by which we are forced to believe in different theories although there are already predictively superior rivals to turn to may be very high, a clear scientific methodology should be able to help us to assure the demarcation between science and pseudoscience. Methods: Karl Popper’s (1902-1994) falsificationist methodology is one of the many approaches to the problem of the demarcation between scientific and non-scientific theories but relies as such too much only on modus tollens and is in fact purely one-eyed. Results: Modus inversus is illustrated in more detail in order to identify non-scientific claims as soon as possible and to help authors not to hide to long behind a lot of self-contradictory and sometimes highly abstract, even mathematical stuff. Conclusions: Modus inversus prevents us from accepting seemingly contradictory theorems or rules in science. Keywords: Science, non-science, modus inversus. E-Mail: Barukcic@t-online.de
Category: Set Theory and Logic

[551] viXra:1910.0443 [pdf] submitted on 2019-10-22 12:47:28

The Real Numbers are Denumerable

Authors: James Edwin Rock
Comments: 2 Pages.

We show Cantor’s diagonal argument has an invalid premise. We create a non-hierarchical Level Set Theory by setting 1/(Aleph Null) = 0. The real numbers have the same cardinality as the set of natural numbers, since the power set of the natural numbers has the same cardinality as the natural numbers. Using binary decimals, we define r, 0 ≤ r ≤ 1 as the ratio of ones to the total number of digits for decimals in the closed interval [0, 1]. We use these ratios to show that the real numbers are denumerable.
Category: Set Theory and Logic

[550] viXra:1910.0418 [pdf] submitted on 2019-10-22 12:15:31

Refutation of Intensional ↓-Operator and Ultra-Filters in Isabelle/ihoml

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate modal collapse as not tautologous and proffer options. On intensional logic using the proof tool Isabelle/IHOML, the following are not tautologous: ↓-operator; δ-ultrafilters; and γ-ultrafilters. In particular, the ↓-operator is not binary (bivalent) as claimed. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[549] viXra:1910.0413 [pdf] submitted on 2019-10-21 21:10:42

Shortest Known Refutation of First and Second-Order Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate for real numbers: 1. In first-order logic the existence of an additive inverse; and 2. In second-order logic the least-upper-bound property, stating every bounded, nonempty set of real numbers has a supremum. Both are not tautologous. This confirms modal propositional and predicate logic. The negative results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[548] viXra:1910.0399 [pdf] submitted on 2019-10-21 10:06:22

Refutation of Second-Order Arithmetic

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

Ignoring set theory and induction as inference, we evaluate arithmetic comprehension (ACA0) and recursive comprehension (RCA0) which are not tautologous. This refutes second-arithmetic and its implications of: recursive mathematics; intermediate value theorem; Turing ideals; recursive sets; weak König’s Lemma; ω-models, Heine-Borel theorem, compactness and completeness of first-order logic; Scott sets; Bolzano-Weierstrass theorem, least upper bound theorem, Ramsey’s theorem for N3; closure under Turing jump, arithmetic sets; and comprehension Cantor-Bendixson theorem, Silver’s theorem, and Kruskal’s theorem. These results for a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[547] viXra:1910.0343 [pdf] submitted on 2019-10-18 14:37:46

Refutation of Hyperfinite Measure-Preserving of Countable Groups with Model Theory

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate a seminal definition and its proposition as not tautologous, hence refuting measure algebra as the core basis of hyperfinite measure-preserving actions of countable groups with model theory, These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[546] viXra:1910.0325 [pdf] submitted on 2019-10-18 09:08:15

Refutation of Cut-Elimination for Intuitionistic Fuzzy Logic Via Nested Sequents

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the inference rules and axioms of the logic system IF, implied as a superset of connexive logic. The inference rule gen is not tautologous, to refute the system at its outset. Four axioms are also not tautologous. What follows is that the seven properties claimed of separation, symmetry, internality, cut-eliminability, subformula property, admissibility of structural rules, and invertibility of rules are also not confirmed. Hence the IF logic system connexive logic in its entirety, and takes with it the conjecture of cut-elimination for intuitionistic fuzzy logic via nested sequents. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[545] viXra:1910.0319 [pdf] submitted on 2019-10-17 21:40:14

Refutation of Relativized Boolean Algebra as Propositional Awareness Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

From the definitions given, excepting three trivial tautologies as rendered, six equations are not tautologous, and one is contradictory. This refutes relativized Boolean algebras and what follows, namely: concrete relativized Boolean algebras; models of propositional logic; awareness frames and awareness models; modal RB’s; powerset MRBAs; and ultrafilter frames. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[544] viXra:1910.0297 [pdf] submitted on 2019-10-17 07:52:20

Refutation of Reverse Derivative Categories

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the definition of an equality for Cartesian differential categories from the differential combinator. It is not tautologous. This refutes reverse derivative categories. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[543] viXra:1910.0295 [pdf] submitted on 2019-10-17 07:58:34

Refutation of Probability Logic and Connexive Logic

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

Six equations as claimed by probability logic are denied, hence refuting connexive logic and its principle. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[542] viXra:1910.0264 [pdf] submitted on 2019-10-15 12:58:10

Refutation of Bounded Residuated Distributive Lattice-Ordered Groupoids (_b_r_d_g_)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate a condition for the universal theory of bounded residuated distributive lattice-ordered groupoids (brdg) as not tautologous. The proposed binary operator ◦ of the residuation property is also not tautologous and refuted as a bivalent operation. These results further deny that the universal theory and the quasi-equational theory of brdg are both decidable and complete; they form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[541] viXra:1910.0238 [pdf] submitted on 2019-10-14 17:15:29

Refutation of Basic Logic (BL)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

Using classical logic we establish the definition of the new connective tensor ⊗ as a theorem. (This is not supposed to be possible from within classical logic as a subset of the conjecture.) Six rules for formation and reflection are not tautologous. This refutes Basic Logic (bL), forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[540] viXra:1910.0229 [pdf] submitted on 2019-10-14 07:57:16

Shorter Refutation of Craig’s Interpolation and Derived Conjectures

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the formula for Craig’s interpolation as not tautologous. This means it is refuted, along with nine conjectures in the following cascade: proving Beth definability; verifying the division of larger problems into smaller problems; representing knowledge by suppression of information; proving intuitionistic and bi-intuitionistic logics, converse modality, tense logic, and extensions with path axioms; and ; and an orthogonality condition for defining duality between interpolates. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[539] viXra:1910.0220 [pdf] submitted on 2019-10-13 23:05:48

Refutation of the Ranked Pentagon Lattice

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the attribute of ranking lattices of ρ(r) ≥ r which is not tautologous. This refutes the conjecture of ranking lattices for extensions of cofinal and mixed (neither end or cofinal). This further disallows theorems derived therefrom in the pentagon lattice, to form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[538] viXra:1910.0145 [pdf] submitted on 2019-10-09 22:15:21

Refutation of Condensable Models of Set Theory by Satisfaction Classes

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the first axiom for satisfaction of classes which is not tautologous. That, as one of four axioms for a conjunction, in turn refutes a Tarski theorem as the basis of satisfaction classes. That in turn refutes the conjecture of condensable models of set theory. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[537] viXra:1910.0119 [pdf] submitted on 2019-10-08 00:59:49

Refutation of Pythagorean Triples

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the definition of Pythagorean triples as not tautologous. This invalidates its subsequent use with Ramsey colorings and the computerized proof using the inference of induction as based thereon. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[536] viXra:1910.0033 [pdf] submitted on 2019-10-05 13:39:51

Refutation of Clausal Language (CL)

Authors: Colin James III
Comments: 2 Pages. .© Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

Three definitions of pairing in natural numbers are evaluated as not tautologous. Refutation of these seminal definitions as a basis of the causal language (CL) also refute extraction proofs of recursive programs. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[535] viXra:1910.0032 [pdf] submitted on 2019-10-05 13:42:39

Refutation of French Smart Building Code Compliance Checking Based on Semantic Rules

Authors: Colin James III
Comments: 2 Pages. .© Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the model of Fig. 1 reducing room classification to regulations ontology as not tautologous. This refutes the conjecture of French smart building code compliance checking based on semantic rules. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[534] viXra:1910.0031 [pdf] submitted on 2019-10-05 13:44:17

Refutation of Complete Intuitionistic Temporal Logics in Topological Dynamics

Authors: Colin James III
Comments: 1 Page. .© Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate intuitionistic temporal logic (ITL) from its last axiom φ→∀φ which is not tautologous and refutes the seminal basis of intuitionistic temporal logics in topological dynamics. Therefore the claim that ITL is proved decidable and complete is not supported. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[533] viXra:1910.0020 [pdf] submitted on 2019-10-01 18:48:11

Refutation of Induction by Numerical Identity of (-1≤C≤1) = (C≤1)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the numerical identity of (-1≤C≤1) = (C≤1). It is confirmed by induction in 1MM iterations not to have a counter example. However, the numerical identity as a logical equation is not tautologous. When mapped as an implication (-1≤C≤1) → (C≤1), it is tautologous. This experiment speaks to the fact that induction when defined as an implication is not tautologous not deduction when defined as an implication is tautologous. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[532] viXra:1909.0634 [pdf] submitted on 2019-09-30 13:31:47

Refutation of Possible/certain Functional Dependencies for Incomplete Data

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate three equations from a continuing example. The last equation is supposed to be contradictory which is not tautologous. This refutes the seminal example, disallows the counter example as proffered, and refutes the conjecture of allowing incomplete information for possible/certain functional dependencies. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[531] viXra:1909.0595 [pdf] submitted on 2019-09-27 11:17:41

Refutation of the Unified Language for Linear and Relational Algebra (Lara)

Authors: Colin James III
Comments: 2 Pages. .© Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate two proofs of Th. 3 of the Lara language as not tautologous. By extension the Lara language as a unification of linear and relational algebra is also refuted. The conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[530] viXra:1909.0569 [pdf] submitted on 2019-09-26 21:59:53

Refutation of Graph Neural Reasoning

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

Definitions of three embedded rules are not tautologous. This refutes the conjecture of proving unsatisfiability by applying neural reasoning to graphs, and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[529] viXra:1909.0568 [pdf] submitted on 2019-09-26 23:36:34

Refutation of the Weisfeiler-Lehman Theorem

Authors: Colin James III
Comments: 1 Page. .© Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the join equation of Weisfeiler-Lehman method as not tautologous, refuting the conjectured theorem and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[528] viXra:1909.0563 [pdf] submitted on 2019-09-25 06:36:10

Refutation of Second Incompleteness Theorem and Well-Foundedness of Jump Hierarchies

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate Gödel’s second incompleteness theorem (equated to well-founded Kripke-frames and the least element principle) and modal Löb’s theorem (equated to induction). Both are not tautologous and converse. Hence to imply well-foundedness, such as to jump hierarchies, is also not tautologous. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[527] viXra:1909.0554 [pdf] submitted on 2019-09-25 10:42:28

Refutation of Hilbert Logic System HG

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate Axiom 1 (A→(B→A)) of the Hilbert system HG as not tautologous. This refutes it and those conjectures to follow: Gödel algebras; Heyting algebras with the prelinearity law; intuitionistic Kripke models; Gödel LG; and modal multi-valued Gödel logic. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[526] viXra:1909.0531 [pdf] submitted on 2019-09-24 08:00:29

A Flaw in Separation Axioms

Authors: Hannes Hutzelmeyer
Comments: 18 Pages.

The author has developed an approach to logics that comprises, but also goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from geometry, number theory, recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers etc. . The conventional treatment of axiomatic set theory (ZFC) is replaced by the abstract calcule sigma so that certain shortcomings can be avoided by the use of Funcish-Mencish language hierarchy: - precise talking about formula strings necessitates a formalized metalanguage - talking about open arities, general tuples, open dimensions of spaces, finite systems of open cardinality and so on necessitates a formalized metalanguage. 'dot dot dot … ' just will not do - the Axiom of Infinity is generalized in order to allow for certain other infinite sets besides the natural number representation according to von Neumann (i.e. general recursion) - the Axioms of Separation is modified as it seems more convenient - there are only enumerably many properties that can be constructed from formula strings, as these are finite strings of characters from a finite alphabet; this should be kept in mind in connection with the Axioms of Replacement - a new look at Cantor's continuum hypothesis in abstract axiomatic set theory leads to the question of so-called basis-incompleteness versus proof-incompleteness - the Axioms of Separation seem to have a flaw; there is a caveat for axiomatic set theory.
Category: Set Theory and Logic

[525] viXra:1909.0522 [pdf] submitted on 2019-09-24 13:33:15

Refutation of Tarski's Quantifier Elimination (Qe) as a Formal Method Technique

Authors: Colin James III
Comments: 3 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the first order polynomial case of Tarski’s quantifier elimination (QE) for recursive induction using definitions of the crucial theorem. QE is not tautologous. This also refutes the conjecture of QE to enhance safety assurance in deep neural networks. These conjectures form non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[524] viXra:1909.0494 [pdf] submitted on 2019-09-23 22:43:34

Refutation of Lindström Theorems for Fragments of First-Order Logic

Authors: Colin James III
Comments: 3 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate three equations for the example property as not tautologous. This refutes virtually all conjectures, such as Lindström theorems for the six upper fragments of logic in Fig. 1, and the two proof approaches claimed as clever. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[523] viXra:1909.0470 [pdf] submitted on 2019-09-23 06:56:53

Refutation of Ceers (Computably Enumerable Equivalence Relations)

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate definitions from two papers of computably enumerable equivalence relations (ceers) as not equivalent, not tautologous, and not computably reducible. This refutes ceers as a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[522] viXra:1909.0469 [pdf] submitted on 2019-09-23 07:00:20

Refutation of Examples of Monotonicity

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate antecedents and consequents of four disjunctions of monotonicity in the system Natural Logic. The conjecture that these claim different interpretations for disjunction is refuted. The claims form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[521] viXra:1909.0431 [pdf] submitted on 2019-09-20 19:11:30

Refutation of Interpretability Logic, and Vaught and Adjunctive Set Theory

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate axioms and inference rules of interpretability logics ILM and TOL with none tautologous, making doubtful the claimed completeness. We then turn to Vaught and adjunctive set theory with neither tautologous. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[520] viXra:1909.0411 [pdf] submitted on 2019-09-19 13:22:46

Refutation of a Modal Logic for Supervised Learning

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate ten conjectures which are not tautologous, and with four as contradictory. This refutes the approach and models of modal logic for supervised learning as a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[519] viXra:1909.0369 [pdf] submitted on 2019-09-17 14:33:15

Resolution in Classical Modal Logic of a Security Barrier Model

Authors: Colin James III
Comments: 3 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate three states of a security barrier example. For the security barrier as equivalent to the road width, the two cases are staying on the road, or staying on the road or going off of the road. Staying on the road is not a theorem. However staying on the road or going off road is a theorem. For the security barrier as larger than the road width, then staying on the road is effectively enforced as a theorem. This evaluation concludes that casting a problem to substructural epistemic resource logic is an unneeded effort for model resolution. The resolution in classical model logic of a security barrier model relegates the authors’ conjectures to non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[518] viXra:1909.0365 [pdf] submitted on 2019-09-17 21:57:03

Refutation of Axiomatizing Logics of Fuzzy Preferences Using Graded Modalities

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the fuzzy preference relation of the assumed ∧-transitivity theorem as not tautologous, relegating it, along with the subsequent minimal modal logics of a finite residuated lattice and the Bulldozed method, as non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[517] viXra:1909.0335 [pdf] submitted on 2019-09-17 01:14:44

Refutation of Modal Operators on Rings of Continuous Functions

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate five expressions for a definition (2), two remarks (2), and a lemma proof (1). None is tautologous. This refutes the titled conjecture which drags in the Hausdorff and Stone topologies, Kripke frames, and dualities of Gelfand-Naimark-Stone and Esakia-Goldblatt. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[516] viXra:1909.0284 [pdf] submitted on 2019-09-13 19:42:35

Refutation of Predicate Transformer Semantics for Hybrid Systems

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the steps of the approach to verify hybrid systems in the style of dynamic logic. Top tier input assumes modal Kleene algebras which are not bivalent. Middle tier processing invokes binary relations (which we do not test). Bottom tier output produces Lipschitz continuous vector fields as verification of hybrid store dynamics, which are not bivalent. This refutes the Isabelle framework for hybrid systems verification, and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[515] viXra:1909.0256 [pdf] submitted on 2019-09-11 11:50:18

Square of Opposition as Meth8 Corrected

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The modern revision of the square of opposition is not tautologous and forms a non tautologous fragment of the universal logic VŁ4. Consequently we redefine the square so that it is validated as tautologous. Instead of definientia using implication for universal terms or conjunction for existential terms, we adopt the equivalent connective for all terms. The modal modifiers necessity and possibility map quantifiers as applying to the entire terms rather than to the antecedent within the terms.
Category: Set Theory and Logic

[514] viXra:1909.0240 [pdf] submitted on 2019-09-10 16:35:12

Refutation of Bitstring and Question-Answer Semantics

Authors: Colin James III
Comments: 3 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate bitstring semantics and its follow-on by partition. Its ordered set of exhaustive predicates is not bivalent but a probabilistic vector space. Its calculus of relations is not tautologous. Hence its broader framework of question-answer semantics (QAS) is not tautologous. The conjecture of “generalizing the Aristotelian square within one common gathering” is denied. What is affirmed is the Meth8 corrected, modern, revised square of opposition is a square, to mean the following conjectures are probabilistic vector spaces: collapsed number line of opposition; non-standard quadrilateral of oppositions; and colored square of oppositions. Bitstring semantics and the extended QAS form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[513] viXra:1909.0237 [pdf] submitted on 2019-09-10 20:36:44

Refutation of Cook-Reckhow Definition

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the Cook-Reckhow definition as represented for conjectures of a polynomial equation and integer linear inequalities with both as not tautologous. Hence we do not evaluate the conjecture of a relational form on which the paper subsequently relies. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[512] viXra:1909.0081 [pdf] submitted on 2019-09-04 19:09:54

Refutation of Automorphism and Borel Dynamics

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the definition of automorphism as not tautologous, hence refuting conjectures of Borel dynamics, and forming non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[511] viXra:1909.0079 [pdf] submitted on 2019-09-04 23:26:42

Refutation of Skolem Witness and Counter-Example Functions

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the Skolem witness function and counter-example function. The functions are not tautologous and not counter-exemplary. This refutes the conjecture of a witness function as opposite a counter-example function and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[510] viXra:1909.0058 [pdf] submitted on 2019-09-03 23:11:03

Refutation of the Russell-Prawitz Embedding

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the Russell-Prawitz embedding as not tautologous. Hence atomization of universal instantiation does not follow (nor does proof reduction, weakening of dinaturality conversion, or strict simulation). These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[509] viXra:1908.0601 [pdf] submitted on 2019-08-29 19:03:58

Refutation of Bell's Theorem for Temporal Logic

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate Bell's theorem for temporal logic. It is not tautologous. Hence schematics of a protocol for a violation of Bell's inequalities for temporal order are similarly moot. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[508] viXra:1908.0580 [pdf] submitted on 2019-08-28 09:52:17

Refutation of Extended Truth Definitions to Peano Arithmetic

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate proof-theoretic analysis by iterated reflection and ordinal analysis of iterated arithmetical comprehension. The former is not tautologous, and the latter is a contradiction. This refutes the extended truth definitions as proffered on Peano arithmetic and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[507] viXra:1908.0572 [pdf] submitted on 2019-08-28 23:35:30

Refutation of Tannakian Categories Via Profinite Groups of Iwasawa Embedding

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the seminal equation to axiomatize profinite groups with the Iwasawa embedding property. It is not tautologous. This taints Tannakian categories and subsequent conjectures to establish model theory of proalgebraic (pro-affine algebraic) groups. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[506] viXra:1908.0554 [pdf] submitted on 2019-08-27 18:38:37

Solution to the Traveling Salesman Problem as a Theorem, Unrelated to P, NP

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the traveling salesman problem (TSP) to confirm it as a theorem but with multiple solutions for n = 4 cities. The number of solutions here is also given by n = 4. Our results do not relate to P, NP, or NP-hard. Hence the salesman problem as an outstanding mathematical problem of optimization is refuted, and as such becomes a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[505] viXra:1908.0449 [pdf] submitted on 2019-08-21 11:27:58

Refutation of Resolution-Based Decision Procedure for Two Variables with Equality

Authors: Colin James III
Comments: 3 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate six equations and three conjectures for the decision procedure. None is tautologous. This refutes the procedure for two variables with equality, and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[504] viXra:1908.0445 [pdf] submitted on 2019-08-21 19:12:23

Availability of Meth8/VŁ4 Demo for 2-Variables (P,q) with Unlimited Sequents

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

This is free on request from info@ersatz-systems.com. Please state name and organization to receive: Unrestricted m8_executable.exe; Instructions.txt with known anomalies; and Editable sample meth8_input_file.txt. The input file contains the shortest confirmation of McCune's proof of Huntington's equation: LETp, q:a, b. (~(~p+q)+~(~p+~q))=p ;TTTT TTTT TTTT TTTT From:Huntington's equation, Robbins algebra, Robbins equation: en.wikipedia.org/wiki/Robbins_algebra ; Dahn, B. I. (1998). Abstract to "Robbins algebras are Boolean: a revision of McCune's computer-generated solution of Robbins problem," Journal of Algebra 208(2): 526–32; Mann, A.L. (2019). A complete proof of the Robbins conjecture. math.colgate.edu/~amann/MA/robbins_complete.pdf ; Robbins algebras are Boolean. calculemus.org/MathUniversalis/4/6robbins.html
Category: Set Theory and Logic

[503] viXra:1908.0442 [pdf] submitted on 2019-08-21 22:59:52

Refutation of Paraconsistent Logic on One Conjecture

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the seminal equivalence and replacement formula of paraconsistent logic, that one formula is equivalent to another in the sense that either can be substituted for the other wherever they appear as a subformula. It is not tautologous, and hence relegates paraconsistent logic to a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[502] viXra:1908.0426 [pdf] submitted on 2019-08-20 14:42:16

Refutation of the Hoop and Pocrim in Heyting Algebras

Authors: Colin James III
Comments: 4 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate 14 equations for the hoop and pocrim in Heyting algebras as not tautologous. The methodology of using proof assistants Prover9 and Mace8 is also refuted. These artifacts form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[501] viXra:1908.0363 [pdf] submitted on 2019-08-17 22:54:44

Scalability of Meth8/VL4

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We offer two more packages in Meth8 to process increasing numbers of propositional variables.
Category: Set Theory and Logic

[500] viXra:1908.0293 [pdf] submitted on 2019-08-15 07:59:10

The Problem P=NP

Authors: Thierry DELORT
Comments: 6 Pages.

In this article, we are going to solve the problem P=NP for a particular kind of problems called basic problems of numerical determination. Nonetheless, this solution can be generalized to all problems belonging to class P or NP. We are going to propose 3 fundamental Axioms permitting to solve the problem P=NP, but those Axioms can also be considered as pure logical assertions, intuitively evident and never contradicted, permitting to understand the solution of the problem P=NP. Indeed, we will see that the conclusion of this article solves the considered problem.
Category: Set Theory and Logic

[499] viXra:1908.0274 [pdf] submitted on 2019-08-14 04:00:14

Refutation of the Paradox of Hempel’s Raven

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the hypothesis which is not tautologous and hence not a paradox. It forms a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[498] viXra:1908.0273 [pdf] submitted on 2019-08-14 05:03:53

Refutation of Differential Reasoning

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We refute Hempel’s raven paradox elsewhere, hence refuting differential reasoning which forms a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[497] viXra:1908.0221 [pdf] submitted on 2019-08-12 06:13:45

Refutation of Perfect and Strong Functions in Fuzzy Logic

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

For fuzzy logic, we evaluate the definition of the residuated lattice operator as not tautologous. We evaluate the perfect fuzzy function and strong (surjective) fuzzy function as not tautologous, but logical equivalents. This refutes the fuzzy functions and distinctions. These form a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[496] viXra:1908.0220 [pdf] submitted on 2019-08-12 06:15:51

Refutation of Zadeh's Swedes and Italians Challenge as a Logic Problem

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Abstract: For Zadeh's Swedes and Italians challenge problem, we evaluate the linguistic weighted average (LWA) for the conjecture that Swedes are on average taller than Italians. None is tautologous, refuting the challenge as a logic problem. These form a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[495] viXra:1908.0219 [pdf] submitted on 2019-08-12 06:16:53

Refutation of Shared Variables in Cross Axiom Models of Alternating Turing Machines

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate shared variables for the reduction of alternating Turing machines (ATMs) to subset space logic (SSL). The purpose of shared variables is to use binary counters for mapping cross-axiom models. None is tautologous, to refute cross-axiom models in the completeness conjectures. These form non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[494] viXra:1908.0211 [pdf] submitted on 2019-08-11 10:48:58

Refutation of the Hrushovski Construction, to Confirm Lachlan and Zil'ber

Authors: Colin James III
Comments: Pages.

A condition for the Hrushovski construction is not tautologous, refuting it. This also denies alleged refutations using it, namely, to confirm the Lachlan conjecture and Zil'ber conjecture. The construction forms a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[493] viXra:1908.0163 [pdf] submitted on 2019-08-09 18:11:53

Refutation of Abductive Repair in Ontology Engineering

Authors: Colin James III
Comments: 3 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the stated example of complete-debug problem (CDP) in formulas framing the definitions, oracles, and repairs. None is tautologous. This forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[492] viXra:1908.0146 [pdf] submitted on 2019-08-08 08:03:40

Refutation of Leibniz' Identity of Indiscernibles and Leo-III Theorem Prover

Authors: Colin James III
Comments: Pages.

Leibniz' identity of indiscernibles as ∀x∀y[∀F(Fx↔Fy)→x=y] is not tautologous. Consequently the Leo-III theorem prover for higher-order paramodulated extensional logic is also refuted. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[491] viXra:1908.0137 [pdf] submitted on 2019-08-07 11:08:28

Refutation of Completeness of Temporal Logics Over Infinite Intervals

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate three pairs of equations for interval temporal logics [ITL]. None is tautologous, refuting completeness of temporal logics for finite and infinite intervals. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[490] viXra:1908.0136 [pdf] submitted on 2019-08-07 11:10:21

Refutation of the Variety of Distributive Bilattices

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The example stated of a distributive bilattice is not tautologous. This refutes that variety and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[489] viXra:1908.0135 [pdf] submitted on 2019-08-07 11:11:24

Refutation of Varieties of Positive Modal Logic (PML)

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Four definitions of positive modal logic (PML) are not tautologous. This refutes positive modal algebra (PML) on the bounded distributive lattice and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[488] viXra:1908.0125 [pdf] submitted on 2019-08-07 21:45:28

Refutation of Saturated Free Algebras (Revisited) and Almost Indiscernible Theories

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate two papers as probabilistic vector spaces with no bivalent basis, non existence, and no meaning. This refutes saturated free algebras and "almost indiscernible theory", forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[487] viXra:1908.0079 [pdf] submitted on 2019-08-04 11:28:10

Refutation of the Term Rewriting Approach to Automated Theorem Proving

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Using three group axioms, two examples for an original and rewritten proofs are not tautologous. This refutes the term rewriting approach for automated theorem proving to form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[486] viXra:1908.0071 [pdf] submitted on 2019-08-05 04:49:12

Refutation of Symmetry Breaking, Boolean Skeletons, Ensemble Technique, and SMT Solver

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The SMT problem as stated is not a theorem, and the derivation of Boolean skeletons, while equivalent, are not the SMT problem. This refutes the symmetry breaking technique and also the attendant ensemble technique. What follows is the SMT solver is refuted, forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[485] viXra:1908.0067 [pdf] submitted on 2019-08-05 06:04:40

Refutation of Classification of Finitary, Algebraizable Logics as Undecidable in Hilbert Calculi

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Two axioms as tested are not tautologous, to refute the Hilbert calculus as claimed. This refutes the conjecture of classification of finitary, algebraizable logic as undecidable in Hilbert calculi. That also disallows a follow-on article. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[484] viXra:1908.0042 [pdf] submitted on 2019-08-02 10:00:13

Refutation of Löwner (Loewner) Order and Quantum Temporal Logic

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the Löwner order ⊑ for positive definite as the basis for quantum temporal logic (QTL). The operator is not tautologous. We also evaluate the semantics for QTL in three operators, also not tautologous. These form non tautologous fragments for both in the universal logic VŁ4.
Category: Set Theory and Logic

[483] viXra:1908.0039 [pdf] submitted on 2019-08-02 18:44:05

Refutation of the Wellfoundedness of the Multiset Order

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The two equations for an inductive proof of the wellfoundedness of the multiset order are not tautologous, and form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[482] viXra:1908.0038 [pdf] submitted on 2019-08-02 19:35:42

Refutation of the Church-Rosser Theorem

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The Church-Rosser theorem evaluates as not tautologous, hence forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[481] viXra:1907.0628 [pdf] submitted on 2019-07-31 07:45:14

Refutation of Completeness for Inclusion and Equivalence of Universality

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate a formula for inclusion and equivalence of universality. It is not tautologous, refuting the conjecture of completeness, and forming a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[480] viXra:1907.0613 [pdf] submitted on 2019-07-30 06:28:12

Refutation of Turing's Halting Problem as Logically Unsolvable

Authors: Colin James III
Comments: Pages.

We confirm the halting conjecture as tautologous and hence refute the halting problem as unsolvable. What follows is that first order logic is decidable. This proof was made possible by the universal logic VL4.
Category: Set Theory and Logic

[479] viXra:1907.0583 [pdf] submitted on 2019-07-29 15:57:59

Refutation of Probabilistic Approximate Logic (Palo) and Logical Imagination Engine

Authors: Colin James III
Comments: 1 Page. .. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

A key property of probabilistic approximate logic (PALO) as one form of inference (of many) is evaluated as not tautologous. This refutes its semantics of the logical imagination engine and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[478] viXra:1907.0582 [pdf] submitted on 2019-07-29 17:17:55

Refutation of Leibniz' Identity of Indiscernibles and Leo-III Theorem Prover

Authors: Colin James III
Comments: Pages. Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com; email: info@cec-services . com

Leibniz' identity of indiscernibles as ∀x∀y[∀F(Fx <-> Fy) -> x=y] is not tautologous. Consequently the Leo-III theorem prover for higher-order paramodulated extensional logic is also refuted. These form a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[477] viXra:1907.0543 [pdf] submitted on 2019-07-27 20:40:04

Refutation of the Algebra of Binary Relations as Basis of Free Kleene Algebras with Domain

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The definitions for composition of relations and set-theoretic union are not tautologous. This refutes the algebra of binary relations on which is based the free Kleene algebras with domain, to form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[476] viXra:1907.0503 [pdf] submitted on 2019-07-25 06:32:58

Refutation of a Formula for Systems of Boolean Polynomials to Parameterized Complexity

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Three formulas defining Boolean polynomial arithmetic are not tautologous, to refute the conjecture. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[475] viXra:1907.0495 [pdf] submitted on 2019-07-25 11:33:12

An Anomaly in the Set of Algebraic Real Numbers

Authors: James Edwin Rock
Comments: 1 Page.

Depending on their representation, the ratio of irrational algebraic numbers to rationals in (0, 1) is either 0.0 or infinite.
Category: Set Theory and Logic

[474] viXra:1907.0453 [pdf] submitted on 2019-07-25 05:47:22

Refutation of the Orthomodular Law

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com; email: info@cec-services . com

We evaluate the orthomodular law x≤y implies y=x∨(y∧x′), with ′ as negation, which is not tautologous. This forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[473] viXra:1907.0417 [pdf] submitted on 2019-07-22 23:57:17

Refutation of the Borel Base and Hull

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate in two equations the Borel base and hull as not tautologous and contradictory, refuting the conjectures and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[472] viXra:1907.0382 [pdf] submitted on 2019-07-19 10:42:33

Refutation of Bounded Homomorphisms and Finitely Generated Fiber Products of Lattices

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate two equations for the standard method for constructing subdirect products, which are not tautologous. Hence the conjecture of bounded homomorphisms and finitely generated fiber products of lattices is refuted. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[471] viXra:1907.0366 [pdf] submitted on 2019-07-18 07:16:12

Refutation of Shallow Embedding in Martin-Löf Type Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In Martin-Löf type theory (MLTT), we evaluate shallow embedding as the following conjecture: “if we add the rewrite rule ∀x. f x (not x) = true, the expression f true false will not be rewritten to true, since it does not rigidly match the not x on the left hand side”. The conjecture is not tautologous, hence refuting shallow embedding in MLTT and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[470] viXra:1907.0252 [pdf] submitted on 2019-07-16 05:37:16

Refutation of Translation of Implicit Logic (Il) to Explicit Logic (El)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

For translation of implicit logic to explicit logic, using intuitionistic and epistemic logic, seven equations are evaluated, with none tautologous. Two refute the recursive translation of “the intuitionistic truth definition into a syntactic recipe”; three refute “key features of intuitionistic logic in modal terms”; and one refutes “the recursion law after knowledge update [as] the basic dynamic equation of hard information” for public announcement logic (PAL). These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[469] viXra:1907.0196 [pdf] submitted on 2019-07-11 17:52:23

Refutation of Hilbert's First Epsilon Theorem in Intuitionistic and Intermediate Logics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

From the universal quantifier shift of (∀xA(x)→B)→∃x(B→A(x)) as not tautologous, the intermediate logic L is refuted, refuting Hilbert’s first epsilon theorem and intuitionistic logic, and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[468] viXra:1907.0175 [pdf] submitted on 2019-07-10 10:02:20

Refutation of Interpretability Logics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Of the non trivial logics for axioms as evaluated, none is tautologous. Hence the interpretability logic IL is refuted, and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[467] viXra:1907.0142 [pdf] submitted on 2019-07-08 07:01:55

Refutation of the Axiom of Dependent Choices in Mice

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The axiom of dependent choices as ∀a∈X∃b∈XP(a, b) ⇒ ∃f: ω → X∀n P(f(n), f(n+1)) is not tautologous. What follows is that the axiom of determinacy is also not tautologous, hence relegating both axioms to a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[466] viXra:1907.0133 [pdf] submitted on 2019-07-08 18:18:44

Refutation of Isabelle/HOL

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Eight of eleven equations are evaluated as not tautologous. This means the rewrite-engine or simplifier tool is not confirmed, the conjunction is not effectively defined by three rules, and other reasoning steps are not expressed similarly, hence refuting Isabelle/HOL. These anomalies form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[465] viXra:1907.0130 [pdf] submitted on 2019-07-08 20:22:52

Refutation of Analytic Choice Principles: Axioms of Choice and Dependent Choice

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The axiom of Γ choice and axiom of Σ11-dependent choice are not tautologous. Therefore an open problem on the Weihrauch degree of parallelization of the Σ11-choice principle on the integers is not solved by using those axioms. These axioms form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[464] viXra:1907.0118 [pdf] submitted on 2019-07-07 09:58:37

Shorter Refutation of Gödel’s Completeness Theorem

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The completeness theorem rendered as (∀x.R(x,x))→(∀x∃y.R(x,y)) is not tautologous. The application of Isabelle/HOL to prove the same also is not tautologous, to invalidate that tool. These demonstrations form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[463] viXra:1907.0103 [pdf] submitted on 2019-07-06 16:38:32

Refutation of the Axiom of Infinity

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the following. Two definitions of the axiom of infinity are both contradictory. Two definitions to extract natural numbers from the infinite are are not equivalent and not tautologous. Therefore the four definitions for the axiom of infinity are refuted. Therefore these form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[462] viXra:1907.0098 [pdf] submitted on 2019-07-05 08:29:54

Refutation of Open-Universe Causal Reasoning

Authors: Colin James III
Comments: Pages.

The proposition and axiom as tested are not tautologous. This does not “validate an intuitive and familiar set of principles about subjunctive conditionals and the relation of causal influence”. This also does not support “an important class of implicit generative models that can plausibly be treated as genuine causal models” or “enable reasoning beyond the propositional level”. The conjecture of open-universe causal reasoning forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[461] viXra:1907.0053 [pdf] submitted on 2019-07-03 10:26:11

Refutation of the Curry-Howard Correspondence

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In the Curry-Howard correspondence, the identity and composition combinators are not tautologous. In fact, the examples result in equivalent truth table values. Further demonstrated is that the instances of Hilbert, lambda, and sequent fragments are also not tautologous with a recent paper rendered moot. These artifacts form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[460] viXra:1907.0048 [pdf] submitted on 2019-07-02 07:31:03

Refutation of a Weak Set Theory H that Proves Its Own Consistency

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

For theory H, the axioms of extensionality and separation are not tautologous. The theorem to prove any instances of the scheme of ε-induction is also not tautologous. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[459] viXra:1907.0040 [pdf] submitted on 2019-07-02 10:21:41

Refutation of the Fixed-Point Property of Self-Proving for Predicate Modal Logics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Refutation of the fixed-point property of self-proving for predicate modal logics
Category: Set Theory and Logic

[458] viXra:1906.0556 [pdf] submitted on 2019-06-30 02:57:53

Confirmation of VŁ4 as Sound

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Logic VŁ4 is defined as a bivalent classical logic that maps tautology correctly and hence is sound. Because paraconsistent, non bivalent, vector logics cannot map non tautology correctly, they are defined as non tautologous fragments of VŁ4 as a universal logic.
Category: Set Theory and Logic

[457] viXra:1906.0543 [pdf] submitted on 2019-06-28 12:10:06

Refutation of Scott’s Existence Axiom in Sheaf Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In two non-tautologous equations we refute Scott’s existence axiom in sheaf theory. Therefore it does not follow that automating free logic is supported by using “modern proof assistants and theorem provers for classical higher-order logic”, such as the showcased tools Isabelle/HOL, Sledgehammer, and Nitpik. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[456] viXra:1906.0536 [pdf] submitted on 2019-06-29 05:45:03

Refutation of the Boone-Rogers Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The Boone-Rogers theorem states that the uniform word problem for the class of all finitely presented groups with solvable word problem is unsolvable. We show that is not tautologous. We further show that a universal, solvable word problem group is tautologous. The former forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[455] viXra:1906.0523 [pdf] submitted on 2019-06-26 09:01:51

An Explicit Mapping of the Real Numbers to the Set of Natural Numbers

Authors: James Edwin Rock
Comments: 1 Page.

By showing that Cantor’s diagonal technique that lists the real numbers does not even work to show the proper cardinality of the rational numbers, we refute his proof technique. To explicitly reference them the real numbers are represented as the limit of partial decimal sums. We show that the real numbers are a denumerable set. We create a new (Level Set Theory), where all infinite sets have the same cardinality as the set of natural numbers.
Category: Set Theory and Logic

[454] viXra:1906.0517 [pdf] submitted on 2019-06-26 11:09:09

Refutation of Hamkins’ Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Hamkins’ theorem claims “every countable model … of set theory embeds into its own constructible universe … x∈y ↔ j(x)∈j(y)”, which is not tautologous, forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[453] viXra:1906.0516 [pdf] submitted on 2019-06-26 11:10:14

Refutation of Keisler’s Ultraproduct Construction

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The ultraproduct construction of Keisler is based on definitions for proper filter in six equations and for ultrafilter in two equations. The definitions are not tautologous. This refutes the ultraproduct construction as “a uniform method of building models of first order theories which has applications in many areas of mathematics.” Claims by other writers to extend Keisler’s construction similarly fail, to form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[452] viXra:1906.0511 [pdf] submitted on 2019-06-26 23:53:01

Refutation of First-Order Proofs Without Syntax

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Proof examples (3) in the introduction, modal combinatorial proofs (1), and rules in Gentzen’s classical sequent calculus (3) are not tautologous. This refutes the conjecture and approach of first-order proofs without syntax, to form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[451] viXra:1906.0499 [pdf] submitted on 2019-06-27 08:02:39

The Existence of Equilibrium States in Dynamic Systems with Attractive Interaction

Authors: Elizabeth Lemeshko
Comments: 4 Pages. Text in Ukrainian. Mohyla Mathematical Journal, Vol 1 (2018) http://mmj.ukma.edu.ua/article/view/152602

Nowadays, science is characterized by needs of the study of various complex processes and phenomena’s. Today’s research of complex and dynamical systems is one of the most advanced ways of research and evolution of the modern world. Models of biology and ecology, physical models, various economic and social models are typical examples of dynamic systems. The concept of an interactive complex system in modern science is a main tool for construction of mathematical models for solving modern civilization problems and development. The dynamical systems approach to conflict is relatively new, but it has beginning in different research fields. Theory of dynamic systems helps us to understand the experiments, build the mathematical model of iterations and examine behavior and relations between opponents, like distribution of resources and territory, population growing etc. This is a challenging problem of finding and achieving a compromise state for opponents on a common territory has different options to define the task and to choose conflict interaction. In 2016, the monograph by V. Koshmanenko where was introduced new approach for dynamic system of conflict that based on interactions of the opponents in the form probability distribution in the disputed area was published. In particular, presented the concept of a complex dynamic system with attractive interaction. The relevance of this research is improving new dynamical system and researching for a new application of abstract models in everyday life. In this paper briefly fundamentals of the theory of dynamical systems described and the theorem on the existence of a equilibrium state in a the new, perspective for research, dynamical system with attractive interaction in terms of probability distributions (measures) and their densities, formulated and proved.
Category: Set Theory and Logic

[450] viXra:1906.0496 [pdf] submitted on 2019-06-27 12:23:34

Refutation of Unification Nets (Canonical Proof Net Quantifiers)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Using the drinker’s paradox, as rendered in two equations, we evaluate the unification net, in two equations, as not tautologous. To extend the unification net to additives is similarly defective, forming a non tautologous fragment of the universal logic VŁ4. We also supply analysis of Smullyan’s drinking principle.
Category: Set Theory and Logic

[449] viXra:1906.0491 [pdf] submitted on 2019-06-25 08:05:42

Refutation of Computer-Simulation Model Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Computer simulation model theory (CSMT), as a substituted extension of mathematical model theory (MMT), is a conjecture that: for a formula φ, construct a computer simulation model S such that 1- φ does not hold in S, and 2- the reasoner I (human being, the one who lives inside the reality) cannot distinguish S from the reality (R), then I cannot prove φ in reality. The conjecture is not tautologous. While we show elsewhere that P=NP is not tautologous (via refutation of the Schaefer theorem), the unprovability of P=NP does not follow from this CSMT approach. This conjecture forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[448] viXra:1906.0479 [pdf] submitted on 2019-06-25 21:05:45

Refutation of Program Verification by Reduction

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The formula for program verification by reduction is not tautologous, thereby refuting such an approach and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[447] viXra:1906.0473 [pdf] submitted on 2019-06-24 07:49:30

Refutation of Kripke Frames from Incompleteness of BAO’s with ◊⊥=⊥

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Because Kripke frames require ◊⊥=⊥, not tautologous, they are refuted. What follows is BAOs so defined are also refuted (which we respectively demonstrate elsewhere), namely: Jónsson-Tarski, Lemmon-Scott; Fine-Thomason, van Benthem, Boolos-Sambin, and Lindenbaum-Tarski. These results also make the Blok dichotomy suspicious. Therefore these conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[446] viXra:1906.0456 [pdf] submitted on 2019-06-23 09:18:35

Refutation of Formalization of Axiom of Choice and Equivalent Theorems Using the Coq Tool

Authors: Colin James III
Comments: Pages.

We evaluate two definitions for maximum and minimal set membership, for the nesting of sets, and the equivalence relations of axiom of choice, Tukey’s lemma, Hausdorff maximal principle, maximal principle, Zermelo’s postulate, Zorn’s lemma, well-ordering theorem. None is tautologous, refuting the claims. The authors conclude: “The whole process of formal proof demonstrates that the Coq-based machine proving of mathematics theorem is highly reliable and rigorous. The formal work of this paper is enough for most applications, especially in set theory, topology and algebra.” We refute those assertions based on the non-bivalent performance of the Coq proof assistant. Therefore, these formalizations and methodology render a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[445] viXra:1906.0448 [pdf] submitted on 2019-06-23 12:06:00

Refutation of Free Choice Permission (FCP) in Deontic Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The formula for free choice permission (FCP) in deontic logic as P(p\/q)→(Pp/\Pq) (FCP) is not tautologous, not a paradox, and hence not applicable in Hilbert-style classical deontic logic as a guarded version. Therefore FCP forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[444] viXra:1906.0407 [pdf] submitted on 2019-06-20 16:58:05

Visualizing the Distributions and Isosurfaces of Some Standard and Non-Standard Quaternion Fractal Sets

Authors: Shawn Halayka
Comments: 16 Pages.

The length, displacement, and magnitude distributions and isosurfaces related to the trajectories of the points in a quaternion fractal set are visualized.
Category: Set Theory and Logic

[443] viXra:1906.0396 [pdf] submitted on 2019-06-21 04:28:02

Refutation of Π1− of Self-Verifying Axioms for Grounding Functions and Relation Predicates

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate Π1− self-verifying axioms defining the grounding functions and the relation predicates as not tautologous. This refutes the approach for weak axiom systems to use subtraction and division primitives, rather than addition and multiplication, to encode formally theorems of arithmetic. Therefore the conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[442] viXra:1906.0380 [pdf] submitted on 2019-06-21 19:35:05

Refutation of Bob Boyer’s Paradox

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Bob Boyer’s paradox is as follows. “A question: It is generally granted that 'p implies p', which is to say, 'if p, then p'. So what about this claim: 'If any number is prime, then any number is prime'?” We find the sentences are unrelated. We then rewrite the second sentence as: “If at least one number is prime, then at least one number is prime”; or as “If at least one number is prime, then possibly all numbers are prime”. Both are trivial tautologies, meaning neither is a contraction or paradox, and therefore forming a tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[441] viXra:1906.0375 [pdf] submitted on 2019-06-22 06:21:16

Refutation of Comorphism of Sites

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The complex equation evaluated is not tautologous, hence refuting the conjecture of comorphism of sites as a functor with a covering lifting property. What follows is that the following are also refuted: surjections, inclusions, localic morphisms, hyperconnected morphisms, and equivalences of toposes. This further relegates category theory of Grothendieck to a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[440] viXra:1906.0366 [pdf] submitted on 2019-06-19 10:31:01

Confirmation of Perez’ Definition of the Conceivable Statement

Authors: Colin James III
Comments: 4 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Of the 16 equations evaluated, 15 are not tautologous. Elsewhere we correctly proved and confirmed most of the conjectures of the author such as refutations of ZFC, Cantor, and Gödel. The definition of a conceivable statement is confirmed as a theorem, with the other equations forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[439] viXra:1906.0356 [pdf] submitted on 2019-06-19 14:06:44

The Continued Fractions and the Real Numbers

Authors: Antoine Balan
Comments: 1 page, written in english

We give a new definition of the real numbers by mean of the continued fractions.
Category: Set Theory and Logic

[438] viXra:1906.0334 [pdf] submitted on 2019-06-19 00:26:41

Refutation of Predicative Collapse and Arithmetical Comprehension

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The four equations evaluated are not tautologous, hence refuting predicative collapsing principles such as arithmetical comprehension. Therefore these form a tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[437] viXra:1906.0319 [pdf] submitted on 2019-06-17 16:18:42

Refutation of the Hol/isabelle Rejection of E.j. Lowe’s Modal Ontological Argument

Authors: Colin James III
Comments: 5 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Of 20 equations evaluated, 16 are not tautologous. This effectively refutes Lowe’s proof, as rendered by the authors. This also invalidates the authors’ rejection of Lowe’s proof due to incompleteness (six of Lowe’s conclusions are dismissed without evaluation) and due to an interactive, trial by error approach to reconstruct Lowe. Therefore an ideal showcase for the computer-assisted interpretive method using HOL/Isabelle failed. These results form another non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[436] viXra:1906.0305 [pdf] submitted on 2019-06-16 12:31:23

Refutation of the Axiomatic Translation Principle for Modal Logic Via Resolution Decision Procedure

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The basis of the axiomatic translation principle is that “no resolution decision procedures are known for decidable first-order logic fragments relevant to modal logics that include the formula ∀xyz(R(x,y)∧R(x,z)→R(y,z)) to express the Euclideanness of a relation R”. We prove a decision procedure for that equation, and hence refute the axiomatic translation principle on which the conjecture is based. Therefore, the axiomatic translation principle is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[435] viXra:1906.0279 [pdf] submitted on 2019-06-15 22:06:36

Rejection of the Quantified Modal Logic Theorem Proving (QMLTP) Library

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate five equations from the quantified modal logic theorem proving (QMLTP) library. None is tautologous for the status of the claimed conjecture, rejecting the approach and library. Other objections include: clarity such as not all problems are in English descriptions; skewed coverage such as about 50% the equations are assumed for Gödel’s embedding; and usability such as the utility tool, to translate QMLTP scripts for pre-selected provers, in Prolog source code which is not compiled into executables for major hardware/OS platforms. Based on these results, the QMLTP approach and library forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[434] viXra:1906.0267 [pdf] submitted on 2019-06-14 08:33:02

Refutation of Inquisitive Modal Logic Via Flatness Grade

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate two seminal definitions for flatness grade with neither tautologous. This refutes inquisitive modal logic and relegates it to a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[433] viXra:1906.0266 [pdf] submitted on 2019-06-14 08:34:23

Refutation of Category Theory by Lattice Identity, and Graph-Theoretic / Set-of-Blocks in Partitions

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate a definition and two models of a method in graph theory to define any Boolean operation. The definition is not tautologous, refuting that only partition tautologies using only the lattice operations correspond to general lattice-theoretic identities. Defined models of graph-theoretic and set-of-blocks do not produce a common edge, but rather show the graph-theoretic definition implies the set-of-blocks definition. This refutes the graph-theoretic model as defining any Boolean operation on lattice partitions of category theory. What follows is that general lattice theory is also refuted via partitions. Therefore the conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[432] viXra:1906.0196 [pdf] submitted on 2019-06-11 07:05:52

The Proof is in the Pudding: an Outline of New Proof of the Existence of God

Authors: Victor Christianto, Robert Neil Boyd
Comments: 6 Pages. This paper has been submitted to a journal. Comments are welcome

Starting with a review of few known arguments to prove the existence of God, we discuss our argument i.e. Nature's order, Pascal's void and Arrow of Time as Neutrosophic triadic to prove the existence of God. The most convincing one is what we call : the proof is in the pudding, i.e. how direct experience with God is the only way to fill everyone's inner void (cf. Pascal). To write shortly, our spiritual inner void can be filled by direct experience with God. This is what we suggest: the proof is in the pudding.
Category: Set Theory and Logic

[431] viXra:1906.0190 [pdf] submitted on 2019-06-11 15:04:27

Понятие множества и его элементов

Authors: Анатолий Вайчунас
Comments: 2 Pages. in Russian. Только e-meil диалог = onli e-meil dialogue

Понятие множества и его элементов полагается, в теории множеств, интуитивно известным и неопределяемо исходным. Однако, оно является обобщаемым на основе таких понятий как многообразие и разнообразие математических предметов.
Category: Set Theory and Logic

[430] viXra:1906.0167 [pdf] submitted on 2019-06-11 00:26:19

Refutation of Two Variants of Noncontingency Operator

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Four axiomatizations of extensions of L(dot-box) over special frames are not tautologous. Those for symmetry and qe&pe are different, but the respective, second-order renditions are equivalent. This refutes the two variants of noncontingency operator. Therefore the conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[429] viXra:1906.0166 [pdf] submitted on 2019-06-11 00:27:28

Refutation of Templates for Converting Counterexamples to Necessitarianism and Into Internalism

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Five equations of counter-examples for two conjectures are not tautologous. Hence use of templates for metaphysical grounding is refuted in contingentism, externalism, internalism, and necessitarianism. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[428] viXra:1906.0159 [pdf] submitted on 2019-06-09 09:14:14

Refutation of Linear Temporal Logic (LTL)

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate two composite modal atoms which turn out to be reductions into single modal operators. This refutes the notion that “new temporal modalities are obtained” and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[427] viXra:1906.0153 [pdf] submitted on 2019-06-09 12:13:58

Refutation of Temporal Logic Via Instant and Interval/period-Based Models of Time

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The basic properties in seven equations are not tautologous. This refutes temporal logic via instant- and interval/period-based models of time and forms it as a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[426] viXra:1906.0140 [pdf] submitted on 2019-06-08 07:06:54

The Logical Content of Inequalities

Authors: Ilija Barukčić
Comments: 28 pages. Copyright © 2019 by Ilija Barukčić, Jever, Germany. All rights reserved. Published by:

Objective. From a theoretical point of view, the demarcation between science and (fantastical) pseudoscience is in necessary for both practical and theoretical reasons. One specific nature of pseudoscience in relation to science and other categories of human reasoning is the resistance to the facts. Methods. Several methods are analyzed which may be of use to prevent that belief is masqueraded genuinely as scientific knowledge. Results. Modus ponens, modus tollens and modus conversus are reanalyzed. Modus sine, logically equivalent to modus ponens is developed. Modus inversus and modus juris are described in detail. Conclusions. In our striving for knowledge, there is still much more scientific work to be done on the demarcation between science and pseudoscience. Keywords: non strict inequality, quantum mechanics, science, pseudoscience
Category: Set Theory and Logic

[425] viXra:1906.0115 [pdf] submitted on 2019-06-07 09:41:06

Refutation of Self-Referential Sentences and Provability: Antimony of the Liar

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The seven definitions evaluated are not tautologous. The first four equations refute the author’s abstract that “a sentence cannot be denominated by p and written as p is not true”. The next three refute that “in a system in which q denominates the sentence q is not provable it is not provable that q is true and not provable”. The net result is refutation of the liar’s antimony as a paradox (contradiction) and concludes that self-referential sentences are in fact provable as not tautologous. These findings provide a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[424] viXra:1906.0095 [pdf] submitted on 2019-06-08 04:19:06

Availability of Meth8/VŁ4

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Manifest of files distributed in current release 2019 June 4 ver 10 (2019.06.04.10)
Category: Set Theory and Logic

[423] viXra:1906.0092 [pdf] submitted on 2019-06-08 05:14:23

Refutation of Conditional Necessitarianism

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The seminal definition of conditional necessitarianism is not tautologous. The spin-off is to deny the following five conjectures, so based thereon: Truthmaker-dependence (TD); Truthbearer-requirement (TB); Aboutness-requirement (AC); and versions of tf named deflationary and inflationary. Therefore these six conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[422] viXra:1906.0055 [pdf] submitted on 2019-06-04 08:22:22

Refutation of the Predicatively Unprovable Termination of the Ackermannian Goodstein Process

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the Goodstein theorem which is not tautologous. This refutes such follow-on as the Ackermanian Goodstein process and conjecture of its predicatively unprovable termination. Therefore that segment is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[421] viXra:1906.0052 [pdf] submitted on 2019-06-04 10:05:37

Refutation of the Cabannas Conjecture of Objectivity

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the 13 atomic equations, with none tautologous. This refutes the Cabannas conjecture of objectivity, forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[420] viXra:1906.0017 [pdf] submitted on 2019-06-02 16:07:32

The Rebuttal of Colin James III's Refutation of Vidamor Cabannas’ Theory of Objectivity

Authors: Vidamor Cabannas
Comments: 11 Pages. More information and comments on Theory of Objectivity and Author Vidamor Cabannas can be found at www.theoryofobjectivity.com

I analyzed the refutation of the Theory of Objectivity performed by Colin James III in vixra.org/abs/1904.0549 and verified that the applied method as well as the values found were correct. However, there is an error in the conclusion, as it refutes the Theory of Objectivity instead of confirming it. The values found in Colin James III's refutation confirm the exact findings of the Theory of Objectivity. Therefore, as the values that were found confirm those presented by the theory, these values are a confirmation and not a refutation of the theory. In other words, the equation N + 1 = n – 1 is tautological, confirming the existence of a geometric entity that occurred before the onset of the universe, which the Theory of Objectivity calls Nothing. However, Nothing is a geometric entity incompatible with the existence of the universe, which forms a contradiction with non-tautological values at the atomic level. This is because in the era of Nothing, there is no space nor any other element other than the geometrical point known as Nothing. There is no reference. This geometric Nothing does not signify absolute zero and has an informative value. The Theory of Objectivity uses a logical and geometric model to demonstrate how the spherical point called Nothing transformed itself into universal space. Thus, it proves that absolute Nothing does not exist.
Category: Set Theory and Logic

[419] viXra:1906.0015 [pdf] submitted on 2019-06-03 03:35:48

Refutation of the Gödel-Mckinsey-Tarski Translation of Intuitionistic Logic (Ipc)

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the Gödel–McKinsey–Tarski [GMT] translation of IPC “by reduction from intuitionistic logic (IPC) using a series of translations”. The equation is not tautologous. This refutes GMT, the method approach using a series of translations, and IPC itself which form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[418] viXra:1906.0014 [pdf] submitted on 2019-06-03 04:37:32

Refutation of Shevenyonov Extension Nary Antropic to Propositional Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Out of 18 equations evaluated, two were trivial theorems, and 16 were not tautologous. This refutes the Shevenyonov extension nary antropic to propositional logic and relegates it to a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[417] viXra:1905.0610 [pdf] submitted on 2019-05-31 10:08:06

Refutation of BF Calculus (And Square Root of Negation)

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Abstract: We evaluate the eight defining equations of the Spencer-Brown system. None is tautologous. This refutes the subsequent primary arithmetic renamed as BF calculus. We previously refuted the Dunn-Belnap 4-valued bilattice as not bivalent and thus non tautologous, so to draw in refinements and extensions by others and apply BF to it compounds the mistakes. Further producing a square root operation on negative 1 is also not tautologous. Spencer-Brown and BF systems subsequently form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[416] viXra:1905.0598 [pdf] submitted on 2019-05-30 11:26:45

Refutation of the Schaefer Theorem for the P, NP Problem (Undecided)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the Schaefer theorem for the P, NP problem by two examples for Graph-SAT(Ψ ). Neither example is tautologous; while claimed to be different, they result in the same truth table values. (The injection of NP-intermediate does not describe our result.) This refutes NP-complete (and P, NP, NP-hard). We also evaluate the P, NP problem as based on P≤NP with the same result. Therefore P, NP, NP-complete, NP-hard, NP-intermediate form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[415] viXra:1905.0590 [pdf] submitted on 2019-05-31 03:08:42

Representation of Processive Functions in Robinson Arithmetic ?

Authors: Hannes Hutzelmeyer
Comments: 10 Pages.

In connection with his so-called incompleteness theorem Gödel discovered the beta-function. The beta-function theorem is important for the representation of recursive functions in the concrete calcule ALPHA of Robinson arithmetic. The other features are composition and minimization of primitive recursive functions. Recursive functions are no part of Robinson arithmetic, but they are representable by certain formulae. The author has developed an approach to logics that comprises, but goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and so on. The concrete calcule LAMBDA of a natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. They include e.g. Ackermann's function and similar constructions. All recursive functions (that are obtained by minimization too) can be represented in LAMBDA . As long as there is no proof that all processive functions are minimitive recursive (recursive but not primitive recursive) one has the problem of representing them in concrete calcule ALPHA of Robinson arithmetic. As long as the challenge of such a proof is not met there is the conjecture that there are calculative functions that are not representable in Robinson arithmetic. An abstract calcule alphakappa of Robinson-Crusoe arithmetic shows that there exists an even weaker adequate arithmetic than Robinson's.
Category: Set Theory and Logic

[414] viXra:1905.0586 [pdf] submitted on 2019-05-29 08:13:43

Refutation of Modal Logics that Bound the Circumference of Transitive Frames

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate six seminal equations, none of which is tautologous. (The author mistakenly labels the Löb axiom as a “fact” as proved by another author.) Therefore modal logics bounding the circumference of transitive frames is refuted and becomes another non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[413] viXra:1905.0583 [pdf] submitted on 2019-05-29 09:28:18

Refutation of the Keisler Measure in Nip Theory

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In NIP theory, the Keisler measure as φ(x)↦μ(φ(x) ∩X)/μ(X)) is not tautologous, relegating it to a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[412] viXra:1905.0577 [pdf] submitted on 2019-05-29 11:24:49

Refutation of Church’s Thesis as a Consistency Property to Fulfill the Minimalist Foundation

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Church’s thesis (CT) is not tautologous as an essential consistency property to fulfill the requirement of the intensional level of a constructive foundation proposed of the minimalist foundation (MF) for constructive mathematics. Therefore, this relegates CT and MF to a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[411] viXra:1905.0554 [pdf] submitted on 2019-05-28 10:54:57

Refutation of Lusin’s Separation Theorem

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

“In descriptive set theory and mathematical logic, Lusin's separation theorem states that if A and B are disjoint analytic subsets of Polish space, then there is a Borel set C in the space such that A ⊆ C and B ∩ C = ∅.” We evaluate two renditions of that equation, both non tautologous, refuting it. Therefore, the separation theorem of Lusin forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[410] viXra:1905.0547 [pdf] submitted on 2019-05-28 14:26:43

Refutation of Gobbay’s Separation Theorem

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The separation theorem of Gobbay takes eight basic cases and four cases for disjunction. None is tautologous. In fact, three groups of the basic cases share unique truth table result values, and one group of the disjunctive cases shares the same truth table result values. This refutes the theorem and adds it as another non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[409] viXra:1905.0524 [pdf] submitted on 2019-05-27 21:13:09

Refutation of a 'concrete' Rauszer Boolean Algebra Generated by a Preorder

Authors: Colin James III
Comments: 5 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

From the 11 equations tested, we refute 13 artifacts: 1. a condition for "an existential quantifier ∃ … on a Boolean algebra"; 2. "a quantifier ∃ as closure operator on B, for which every open element is closed"; 3. the interior operator on abstract topological Boolean algebra; 4. the kernel of a homomorphism from a Heyting algebra into another as a filter; 5. deductive systems and filters as equivalent; 6. the atomic definition of p ≤ ∃p in Halmos algebra; 7. a ‘concrete’ Rauszer Boolean algebra; 8. two conditions for the definition of a filter (and Heyting algebra using the filter); 9. a De Morgan algebra as a Kleene algebra; 10. equivalences of symmetrical Heyting algebras; 11. equivalences in Heyting algebras; 12. intuitionistic implication of intuitionistic logic; and 13. a theorem and a proposition of Nelson algebras. As a result, the following seven areas are non tautologous fragments of the universal logic VŁ4: 1. Topological Boolean algebra; 2. Heyting algebra; 3. Intuitionistic logic; 4. Halmos algebra; 5. Rauszer algebra; 6. Kleene algebra; and 7. Nelson algebra.
Category: Set Theory and Logic

[408] viXra:1905.0519 [pdf] submitted on 2019-05-28 03:49:06

Refutation of Presburger Arithmetic Via Axiom 2

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In Presburger arithmetic, Axiom 2 as x+1 = y+1 → x=y is not tautologous. Therefore Presburger arithmetic is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[407] viXra:1905.0482 [pdf] submitted on 2019-05-25 06:15:15

Refutation of Bourbaki’s Fixed Point Theorem and the Axiom of Choice

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate Moroianu’s and the Tarski-Bourbaki fixed point theorem and axiom of choice (AC). Two versions of the theorem and then seven theorems and corollary which follow are also not tautologous. Therefore these conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[406] viXra:1905.0477 [pdf] submitted on 2019-05-23 08:12:13

Refutation of Bounded and Σ1 Formulas in pa

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

A fundamental proposition for bounded and Σ1 formulas in PA is not tautologous. While the author states that the informal notes are full of errors, this fundamental mistake causes the entire section about Rosser’s form of Gödel’s theorems to collapse. Therefore the proposition is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[405] viXra:1905.0476 [pdf] submitted on 2019-05-23 08:13:29

Refutation of Logically-Consistent Hypothesis Testing and the Hexagon of Oppositions

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Definitions for ◻H and ¬ ◊H are supposed to be equivalent for a classical mapping of agnostic hypothesis tests. While each definition reduces to a theorem in the conjecture, they are not tautologous. This refutes that agnostic hypothesis tests are proved to be logically consistent. Hence the characterization of credal modalities in agnostic hypothesis tests cannot be mapped to the hexagon of oppositions to explain the logical relations between these modalities. Therefore the 11 definitions tested form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[404] viXra:1905.0475 [pdf] submitted on 2019-05-23 08:14:35

Refutation of Reversing the Counterfactual Analysis of Causation

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The seminal formula of C causes E iff (~C □→ ~E) is not tautologous, that is, it is not a theorem, from which the conjecture is derived. Hence reversing the counterfactual analysis of causation is refuted. Therefore the conjecture forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[403] viXra:1905.0469 [pdf] submitted on 2019-05-23 17:41:36

Refutation of Condition/decision Duality and the Internal Logic of Extensive Restriction Categories

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The equations for condition/decision duality are not tautologous, hence refuting what follows as internal logic of extensive restriction categories. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[402] viXra:1905.0440 [pdf] submitted on 2019-05-22 12:02:07

Refutation that Strawson’s Presupposition is Different from Russell’s Entailment

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Strawson’s presupposition and Russell’s entailment are of the same form, equivalent, and hence not different. These conjectures form a tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[401] viXra:1905.0437 [pdf] submitted on 2019-05-22 18:19:33

Refutation of Replacing Classical Logic with Free Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluated 12 equations for the assertions with none tautologous. Therefore this conjecture is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[400] viXra:1905.0434 [pdf] submitted on 2019-05-22 21:44:39

Refutation of Rooted Hypersequent Calculus for Modal Logic S5

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate two example equations as not tautologous, thereby refuting the rooted hypersequent calculus for modal propositional logic S5. The sequent calculus forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[399] viXra:1905.0358 [pdf] submitted on 2019-05-20 01:17:30

Proposal of Replacing Classical Logic with Free Logic for Reasoning with Non-Referring Names in Ordinary Discourse

Authors: Bornali Paul
Comments: 9 Pages.

Reasoning carried out in ordinary language, can not avoid using non-referring names if occasion arises. Semantics of classical logic does not fit well for dealing with sentences with non-referring names of the language. The principle of bivalence does not allow any third truth-value, it does not allow truth-value gap also. The outcome is an ad hoc stipulation that no names should be referentless. The aim of this paper is to evaluate how far free logic with supervaluational semantics is appropriate for dealing with the problems of non-referring names used in sentences of ordinary language, at the cost of validity of some of the classical logical theses/ principles.
Category: Set Theory and Logic

[398] viXra:1905.0310 [pdf] submitted on 2019-05-19 03:48:48

Refutation of Tarski’s Geometric Axioms and Betweenness

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Of 13 equations evaluated, five are tautologous and eight are non tautologous. This refutes Tarski’s geometric axioms and betweenness which form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[397] viXra:1905.0276 [pdf] submitted on 2019-05-17 09:44:55

Refutation of Counterpart Theory Via Modal Logic

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Of seven examples for counterpart theory, none is tautologous. In fact, a translation is not tautologous in the counterpart model or in QMT, but rather shares the same truth table result. Two definitions of intensionality are also not tautologous and hence refuted. Therefore counterpart theory forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[396] viXra:1905.0275 [pdf] submitted on 2019-05-17 10:56:36

Programming Primitive Recursive Functions and Beyond

Authors: Hannes Hutzelmeyer
Comments: 17 Pages. The details given here relate to document viXra: 1905.0221

In addition to the publication 'The Snark, a counterexample for Church's thesis?' examples and details are offered in the form of two appendices C6 and C7 that allow for better understanding of the general method and the particular problem related to Church's thesis. The author has developed an approach to logics that comprises, but also goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers etc. The most usual approach to calculative (effectively calculable) functions is done by register machines or similar storage-based computers like the Abacus or Turing machines. Another usual approach to computable functions is to start with primitive recursive functions. However, one has to find a way to put this into a form that does not rely on a pre-knowledge about functions and higher logic. The concrete calcule LAMBDA of decimal pinitive arithmetic allows for such an access. It is based on a machine that is completely different from the storage-based machines: the PINITOR does not use storages but rather many microprocessors, one for each appearance of a command in the code of primitive recursive functior. the codes are decimal numbers, called pinons , where only the characters 0 1 2 8 9 appear. There are four kind of commands only: 0 nullification, 1 succession, 2 straight recursion and 8 composition. The PINITOR is a calculator which means that there is no halting problem. Computers have halting problems, per defintion calculators do not. Appendix C6 gives the programming of the codes of most of the usual primitive function and goes even farther, e.g. it introduces generator technique that allows for the straight-forward calculation of so-called processive function, that are not primitive recursive. The most famous examples of Ackermann and other hyperexponential functions are programmed. Appendix C7 turns to the Boojum-function and the Snark-function that have been introduced as calculative functions in the above publication in connection with Church's thesis. A list is provided that gives the lowest values for these functions and gives some more insight into these functions.
Category: Set Theory and Logic

[395] viXra:1905.0254 [pdf] submitted on 2019-05-16 08:59:09

Refutation of Forcing to Change Large Cardinal Strength

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The seminal theorem of the dissertation states a cardinal is greatly inaccessible if and only if it is Mahlo. Three non trivial equations of the proof are not tautologous, thereby refuting theorems derived therefrom. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[394] viXra:1905.0231 [pdf] submitted on 2019-05-15 08:39:24

A Methodology for Urban Sustainability Indicator Design

Authors: Ricardo Alvira
Comments: 19 Pages.

In recent times we have witnessed proliferation of indicators and models for measuring sustainability. This reveals the lack of common and shared scientific paradigm/common framework from which to confront the issue of quantitatively assessing the sustainability of our society. With the aim of moving forward the definition of such common framework, in this article we explain an easy formal methodology for designing urban sustainability indicators based on Fuzzy Logic / Fuzzy Sets Theory. The interest of this methodology is threefold: Firstly, formal procedures enable easier testing, a most fundamental issue forgotten in many current proposals of sustainability indicators. Secondly, a formal procedure is easily understandable and can become a common language allowing shared use of the indicators and facilitating their continuous improvement. And thirdly, fuzzy logic is widely used in computing and artificial intelligence, thus facilitating the progressive automation of our sustainability monitoring models. To help understand the procedure, the design of two indicators is reviewed.
Category: Set Theory and Logic

[393] viXra:1905.0223 [pdf] submitted on 2019-05-15 18:33:17

Refutation of Jaccard Index

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The definition of the Jaccard index is not tautologous, hence refuting it with derivations and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[392] viXra:1905.0221 [pdf] submitted on 2019-05-16 03:15:48

The Snark, a Counterexample for Church's Thesis ?

Authors: Hannes Hutzelmeyer
Comments: 7 Pages.

In 1936 Alonzo Church put forward his thesis that recursive functions comprise all effectively calculative functions. Whereas recursive functions are precisely defined, effectively calculative functions cannot be defined with a rigor that is requested by mathematicians. There has been a considerable amount of talking about the plausibility of Church's thesis, however, this is not relevant for a strict mathematical analysis. The only way to end the discussion is obtained by a counterexample. The author has developed an approach to logics that comprises, but goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and so on. The concrete calcule LAMBDA of natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. All recursive functions can be represented in LAMBDA too. The unary Snark-function is defined by a diagonalization procedure such that it can be calculated in a finite number of steps. However, this calculative function transcends the compinitive functions and presumably the recursive functions. The defenders of Church's thesis are challenged to show that the Snark-function is recursive. Another challenge asks for an example of a recursive function that cannot be expressed as a compinitive function, i.e. without minimization.
Category: Set Theory and Logic

[391] viXra:1905.0215 [pdf] submitted on 2019-05-14 07:54:57

Refutation of Intutionistic Zermelo-Fraenkel Set Theory (Izf), de Jongh’s Theorem, and CZF

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the nine axioms for intutionistic Zermelo-Fraenkel set theory (IZF). None is tautologous. This refutes IZF and its use in blended models and denies De Jongh’s classical theorem and similar results for constructive ZF (CZF). These segments form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[390] viXra:1905.0195 [pdf] submitted on 2019-05-13 12:10:48

Refutation of Reverse Mathematics and Nets

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate four definitions for reverse mathematics (3) and nets (1). None is tautologous. This refutes reverse mathematics and nets. Therefore these definitions form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[389] viXra:1905.0172 [pdf] submitted on 2019-05-12 23:34:42

Refutation of the Barwise Compactness Theorem Via Sublanguage la

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

A definition with variant to establish a sublanguage in support of the Barwise compactness theorem is not tautologous. By extension the theorem is also refuted. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[388] viXra:1905.0160 [pdf] submitted on 2019-05-11 19:27:34

Confirmation of VŁ4 as Complete

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Logic VŁ4 is defined as a bivalent classical logic that maps quantifiers to modalities as a tautology making VŁ4 complete. Paraconsistent, non bivalent, vector logics are defined as non tautologous fragments of VŁ4 as a universal logic.
Category: Set Theory and Logic

[387] viXra:1905.0153 [pdf] submitted on 2019-05-10 18:37:20

The Thot-Queen Paradox

Authors: Analytic Birb
Comments: 2 Pages.

The apparent contradiction of the status of women as both thots and queens, as expressed by the sentences ‘if she breathes, she’s a thot’ and ‘all women are queens’ is examined. The problem is formalized using the law of the excluded middle and a counterexample to the assertion is provided. The implications of the lack of a paradox are discussed.
Category: Set Theory and Logic

[386] viXra:1905.0116 [pdf] submitted on 2019-05-07 15:52:46

Refutation of Pascal’s Wager

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The antecedent and consequent of the thought experiment of Pascal’s wager are not tautologous. However, to determine gain by one wager or the other is tautologous. This refutes the conjecture of Pascal’s wager as ultimately not allowing reason to determine faith. In other words, the “existence of God is possible to prove by human reason”. What follows furthermore is that the existence of God is more profitable from this thought experiment. Therefore the conjecture forms a tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[385] viXra:1905.0097 [pdf] submitted on 2019-05-06 21:21:49

Refutation of a Fast Algorithm for Network Forecasting Time Series from Definition of Visibility Graph

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a definition of the visibility graph as not tautologous to deny a fast algorithm for forecasting time series. Hence the conjecture of a forecasting algorithm is denied. This forms a non tautologous fragment of the universal logic VŁ4. However, we resuscitate the conjecture using the Kanban cell neuron network (KCNN), a linear step-wise function, for the desired conjecture without injected data.
Category: Set Theory and Logic

[384] viXra:1905.0063 [pdf] submitted on 2019-05-04 23:23:21

Refutation of Provability Logic GL and Japaridze’s Derived Polymodal (GLP)

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate eight equations for provability logic (GL) and the derived polymodal logic of Japaridze (GLB, GLP). None is tautologous, hence refuting provability logic. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[383] viXra:1905.0062 [pdf] submitted on 2019-05-04 23:24:52

Refutation of Non Sahlqvist Formulas by Three Counter Examples

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate three equations as examples of non Sahlqvist formulas. None is tautologous. What follows is that Fine’s theorem and monotonic modal logic are refuted. Therefore those conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[382] viXra:1905.0051 [pdf] submitted on 2019-05-03 10:27:21

Refutation of Isabelle / H O L Prover Assistant

Authors: Colin James III
Comments: 1 Page. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com; email: info@cec-services . com

A meta-rule for structural induction in the prover assistant Isabelle/HOL is not tautologous. This refutes the assistant and denies it can effect a cross-fertilization of computer science and metaphysics. Therefore Isabelle/HOL is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[381] viXra:1905.0038 [pdf] submitted on 2019-05-02 13:39:14

Refutation of the Language of Sets for Model Theory = Universal Algebra + Mathematical Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

A first order language of sets is proposed, but the first example A⊂B iff (x∈A then x∈B) is not tautologous. This refutes the conjecture of model theory = universal algebra + mathematical logic, which forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[380] viXra:1905.0015 [pdf] submitted on 2019-05-01 11:33:18

Refutation Arrow’s Impossibility Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two versions of Arrow’s impossibility theorem with disjunctive or conjunctive results. Both as rendered are not tautologous. This means Arrow’s framework is refuted, hence coloring the conjecture of Arrow’s theorem before pivotal voters or dictators can be derived. Therefore Arrow’s impossibility theorem forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[379] viXra:1905.0011 [pdf] submitted on 2019-05-01 17:44:36

Refutation of Overlap Algebras Constructively to Prove Complete Boolean Algebras

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate four equations which are not tautologous, but in fact produce the equivalent logic table values result. This means that the stated problem of applying singletons to the powerset is equivalent to proving singletons are atoms and that every subset satisfying a singleton is also an atom. Hence, overlap algebras do not constructively prove complete Boolean algebras. Therefore that conjecture for intuitionistic logic forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[378] viXra:1904.0549 [pdf] submitted on 2019-04-28 17:05:57

Refutation of Cabannas Theory of Objectivity

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate equations about adding or subtracting something from nothing. The duals as a disjunction are tautologous. However that disjunction is not itself equivalent to nothing. This refutes the Cabannas theory of objectivity at its atomic level, forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[377] viXra:1904.0526 [pdf] submitted on 2019-04-27 23:38:01

Refutation of Remainder Sets for Paraconsistent Revisions

Authors: Colin James III
Comments: Pages.

Two definitions for expansion, remainder, and selection of K functions are not tautologous. Two definitions implication and paraconsistent/weak negation operators are not tautologous. These refute remainder sets and paraconsistent valuations of logic mbC, an extension of CPL+. Therefore these conjectures are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[376] viXra:1904.0516 [pdf] submitted on 2019-04-26 10:24:51

Refutation of Collection Theory as the Set of Universal Closure of Sentences

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate collection theory as the set of universal closure of sentences in a schema equation. It is not tautologous. This refutes Collection as the conjectured schema. Therefore collection theory is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[375] viXra:1904.0510 [pdf] submitted on 2019-04-26 19:46:15

Refutation of Ishihara’s Tricks and (Seemingly) Impossible Theorems in Constructive Mathematics

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

The precise definition of LPO and Ishihara’s tricks as rendered in four equations are not tautologous. This refutes LPO and Ishihara’s tricks. What follows is that (seemingly) impossible theorems in constructive mathematics are denied as theorems. Therefore those conjectures are non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[374] viXra:1904.0493 [pdf] submitted on 2019-04-25 13:17:26

Refutation of Ordinal Notation Via Simultaneous Definition

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate five definitions as not tautologous. This refutes the conjecture that inductive-recursive definitions can give rise to ordinal notation systems that uniquely represent ordinals. Hence the definitions and conjecture are non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[373] viXra:1904.0484 [pdf] submitted on 2019-04-24 06:38:20

Riemann's Hypothesis (The Millenium's Problem)

Authors: Louiz Akram
Comments: 3 Pages.

This is a better scan of my work about riemann's hypothesis , I proved that there are no zeros when Re(s)<1 . I didn't suppose that zeta is convergent , but I supposed that the zero is among the images with the relation zeta of a known s=a+ib since zeta is a relation when it doesn't converge. There is an issue when we try to find an extension for Riemann's zêta : The complex form multiplied to zêta should never be null , because that form should also devide zêta to deduce the extension. The mistake is to multiply by a complex form then devide by only an image by it. There is also the logical problem when zêta which is divergent equals a convergent extension. I think that suppositions should be made before trying to find an extension to Zeta.
Category: Set Theory and Logic

[372] viXra:1904.0483 [pdf] submitted on 2019-04-24 06:51:32

Refutation of Domain Theory and the Scott Model of Language PCF in Univalent Type Theory

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

The definitions of directed, complete posets for antisymmetry and transitivity are not tautologous, thereby refuting basic domain theory. By extension, the Scott model of language PCF in univalent type theory is also refuted and another non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[371] viXra:1904.0466 [pdf] submitted on 2019-04-24 12:51:12

Refutation of Strong Jump Inversion and Decidable Copy of a Saturated Model of DCF0

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the paper’s abstract, the definition of strong jump inversion is not tautologous, hence strong jump inversion is refuted. A computable enumeration of the types realized in models of DCF0 is also refuted. The alleged fact that the saturated model of DCF0 has a decidable copy is denied. Therefore these conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[370] viXra:1904.0404 [pdf] submitted on 2019-04-20 08:15:10

Refutation of Type Theory

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the subset equations for the basis of type theory. They are not tautologous. Therefore the basis of type theory is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[369] viXra:1904.0385 [pdf] submitted on 2019-04-19 12:16:13

Refutation of logic first order team (FOT)

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a definition equation as not tautologous, hence refuting first order team (FOT) logics. Therefore FOT is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[368] viXra:1904.0369 [pdf] submitted on 2019-04-18 07:25:21

Refutation of Huemer’s Confirmation Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

Huemer proposed a confirmation theory to solve the problem of induction. In lieu of the excess of content from Popper and Miller, the proposed replacement is also not tautologous, so we correct it for the intended use. That applied to the subsequent proposal in three parts shows one part is not tautologous, hence denying the proposal. Therefore Huemer’s confirmation theory is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[367] viXra:1904.0365 [pdf] submitted on 2019-04-18 08:26:57

Refutation of Parikh’s Axiomatization of Game Logic G and Completeness of Logic System Par

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

Three equations of Parikh’s axiomatization of game logic G are not tautologous. Hence, the extended logic system Par is refuted, and Parikh’s completeness conjecture is also refuted. Therefore these artifacts are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[366] viXra:1904.0312 [pdf] submitted on 2019-04-16 17:41:49

Confirmation of the Collapse of the Buss Hierarchy of Bounded Arithmetics

Authors: Colin James III
Comments: Pages.

Two seminal rules of inference evaluated as not tautologous. This means the following are also refuted: Buss’s hierarchy of bounded arithmetics does not entirely collapse; Takeuti’s argument implies P ≠ NP; and systems PV and PV−. What follows is that separation of bounded arithmetic using a consistency statement is not viable. Therefore the above are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[365] viXra:1904.0311 [pdf] submitted on 2019-04-16 22:13:42

Confirmation that Impossible Worlds Mean Nothing is Necessary there But Everything is Possible.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We confirm the definition that impossible worlds mean “nothing is necessary there but everything is possible”. The truth axiom as given is not tautologous, but rather the logical value of truthity. The rules of extensionality and monotonicity as given are not tautologous. When the definition of impossible worlds is combined with monotonicity + T + 4 that conjecture is also tautologous. Therefore the failed equations are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[364] viXra:1904.0292 [pdf] submitted on 2019-04-15 10:26:51

Refutation of Ramsey’s Theorem Via Pythagorean Triple of Integers

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the lemma proffered to prove Ramsey’s theorem, and the strengthened lemma in a note. Neither are tautologous. We evaluate the proof of Pythagorean triple of integers as colored. It also is not tautologous. In fact, the coloring or non-coloring produces a logically equivalent result, meaning the Ramsey theorem is neither a tautology nor a contradiction. This implies “the inductive hypothesis” is suspicious. What follows is that the HOL proof assistant for the Ramsey theorem is an historical enormity in its 200 TB computer program with a certified prize result. Therefore the Ramsey theorem and HOL proof assistants are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[363] viXra:1904.0290 [pdf] submitted on 2019-04-15 11:33:56

Refutation of Inclusion Logic

Authors: Colin James III
Comments: Pages.

We evaluate a seminal equation from a proof sketch, which is not tautologous. By extension, this means dependence logic, inclusion logic, and independence logic are also not tautologous. Therefore dependence logic, inclusion logic, and independence logic are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[362] viXra:1904.0284 [pdf] submitted on 2019-04-15 20:35:54

Refutation of Ethical Reasoning and Hol as a Universal Meta-Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

An exemplary equation in HOL for ethical reasoning is not tautologous. By extension, HOL is refuted as “a universal meta-logic”, and “ethical reasoning” is refuted. Therefore HOL and ethical reasoning are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[361] viXra:1904.0182 [pdf] submitted on 2019-04-09 14:11:05

Refutation of Fodor’s Causality Principle and Extension to the Class Fodor Principle

Authors: Colin James III
Comments: Pages.

We evaluate Fodor’s principle in two versions we name weaker and stronger. Neither is tautologous. By extension, the class Fodor principle is not tautologous. Because of this, using Kelly-Morse set theory as a basis for denial is rendered moot. However, to rely on KM set theory to deny Fodor principles as a class principle is obviated by the refutation above. Therefore Fodor’s causality principles and extension are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[360] viXra:1904.0164 [pdf] submitted on 2019-04-08 19:37:00

Refutation of Temporal Type Theory (TTT), Temporal Landscapes, and Scott’s Topology

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From a footnote, and then a three part definition, temporal type theory (TTT), and then temporal landscapes for open sets are not tautologous and hence refuted. That further refutes Scott’s topology. These results therefore form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[359] viXra:1904.0148 [pdf] submitted on 2019-04-07 09:55:15

Student Quiz Denied as a Paradox and Refuted as a Conjecture

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We assume the teacher is veracious and evaluate the assertion “There is possibly a quiz next week on Monday, Tuesday, or Wednesday.” It is not tautologous, hence denying it is a paradox and refuting it as a conjecture. Therefore the student quiz paradox is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[358] viXra:1904.0145 [pdf] submitted on 2019-04-07 17:06:59

“Fuzzy Time”, a Solution of Unexpected Hanging Paradox (a Fuzzy Interpretation of Quantum Mechanics)

Authors: Farzad Didehvar
Comments: 6 Pages.

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzifying Turing Machines but seemingly there is no need to fuzzify time. Throughout this article, we try to change this picture and show why it is helpful to consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Paraconsistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time.
Category: Set Theory and Logic

[357] viXra:1904.0129 [pdf] submitted on 2019-04-06 14:00:21

The Surprise Exam Paradox: Students Should be Surprised on Wednesday or Tuesday.

Authors: Tomonori Hirasa
Comments: 2 Pages.

The students in the surprise exam story reasoned that no surprise exam could take place on any day of the week. Actually, however, the students were surprised on Wednesday by the teacher's surprise exam. In this paper, we show where the students' reasoning went wrong and that students should be surprised on Wednesday or Tuesday.
Category: Set Theory and Logic

[356] viXra:1904.0106 [pdf] submitted on 2019-04-05 23:10:51

Denial of Summers’ Malice and Alice as a Logical Puzzle

Authors: Colin James III
Comments: Pages.

The clauses of the Malice and Alice logical puzzle of Summers are mapped for evaluation of the conjecture. It should return tautology for all pairwise answers, before removing pairs of the antecedent to discover the correct pair of murder and victim. However, the conjecture is not tautologous, albeit one value shy. This means the puzzle as rendered is not well formed, thereby denying the status of a puzzle. Therefore the conjecture is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[355] viXra:1904.0101 [pdf] submitted on 2019-04-04 06:03:04

Refutation of the Axiom of Dependent Choices (DC) on Supercompactness of ω1

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

The axiom of dependent choices (DC) is evaluated in two equations on supercompactness of ω1, with neither tautologous and hence refuting DC. Therefore DC equations are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[354] viXra:1904.0095 [pdf] submitted on 2019-04-04 08:40:19

Propositional Logic Without The Deduction Theorem

Authors: Henry Wong
Comments: 3 Pages.

In propositional logic, given a set of axioms, we can derive formulas. Here we present the derivations of some formulas without the use of the Deduction Theorem. The derivations are presented compactly with only few referrals to other theorems. Most textbooks in this subject avoid this kind of approach.
Category: Set Theory and Logic

[353] viXra:1904.0057 [pdf] submitted on 2019-04-03 15:30:21

Refutation of the Correspondence Theory of Sahlqvist

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the example of mapping the Sahlqvist formula of p ∧ ◊p → ◻p into corresponding quantified expressions. The formula is not a theorem, but the corresponding quantified expressions are theorems. Hence the mapping refutes the Sahlqvist correspondence theory. Therefore these failures are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[352] viXra:1904.0050 [pdf] submitted on 2019-04-03 21:49:58

Refutation of Skolem Axiom Form

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

"A Skolem axiom has the form ∀x,y(φ(x, y)→φ(x,f(x))), where f is a new function symbol introduced to denote a "Skolem function" for φ." The Skolem axiom form is not tautologous, hence refuting it. Therefore the Skolem axiom form is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[351] viXra:1904.0043 [pdf] submitted on 2019-04-02 10:36:44

Refutation of Projective Determinancy Via the Ultrapower

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the definition of ultrapower as a convention. The two states equated to 1 as designated proof value and as ordinal value are not tautologous. This refutes the ultrapower and hence colors the subsequent exposition to deny projective determinancy. Therefore the ultrapower and projective determinancy are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[350] viXra:1904.0027 [pdf] submitted on 2019-04-02 22:53:17

Refutation of Zermelo–Fraenkel (ZF), Method of Forcing, and Continuum Hypothesis (CH)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the fundamental axiom of ZF (or ZFC), ZF is proved contradictory. This means the method of forcing and the continuum hypothesis are also denied. Therefore these results are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[349] viXra:1904.0006 [pdf] submitted on 2019-04-02 02:07:58

Refutation of Ackermann’s Approach for Modal Logic and Second-Order Quantification Reduction

Authors: Colin James III
Comments: 4 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From one source we evaluate the Ackermann rule and from another source three examples in 15 equations of second-order reduction. None of the equations is tautologous. This implies these approaches to map modal clauses of first-order logic to and from second-order logic are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[348] viXra:1903.0510 [pdf] submitted on 2019-03-28 19:36:20

Refutation of Constructive Proof of Craig’s Interpolation Theorem Using Maehara’s Technique

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a constructive proof of Craig’s interpolation theorem by way of Maehara’s technique. Six equations are not tautologous, and serve as antecedents for respective conclusions of two- or four-sequents. Hence, the concluding consequents in any state of proof value will always return a tautology. This means the technique of Maehara does not produce a constructive proof of Craig's interpolation theory as applied to sequent logic for interpolation of non-normal logics. Therefore the approach forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[347] viXra:1903.0494 [pdf] submitted on 2019-03-27 11:41:43

Solution of Horty's Puzzles in STIT Logic

Authors: Colin James III
Comments: Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

In see-to-it-that logic (stit logic), three deontic examples are presented of Horty's coin betting puzzle with two agents. The form of the examples is tautologous. However, a profitability analysis by contrasting outcome for the agents shows none is tautologous. The example for the agent initiating the state of the coin as more profitable than the other agent is more closely aligned to tautology and hence the more profitable strategic outcome. What follows is that stit logic is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[346] viXra:1903.0472 [pdf] submitted on 2019-03-26 15:29:13

Refutation of Dyatic Semantics on Paraconsistent Logic C1

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the example of applying dyatic semantics to paraconsistent logic C1, four axioms are not tautologous, hence refuting that approach. Therefore paraconsistent logics are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[345] viXra:1903.0471 [pdf] submitted on 2019-03-26 15:31:55

Refutation of the Gödel Class with Identity as un-Solvable

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the Gödel quantification scheme of ∃*∀n∃*φ as tautologous. When it is extended to decidable and undecidable prefix-classes, none is tautologous. This refutes the Gödel class with identity as undecidable, to mean it is in fact solvable as not tautologous. Therefore the prefix-classes are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[344] viXra:1903.0461 [pdf] submitted on 2019-03-27 05:43:26

On the Way to a Foundation of Mathematics, Version 7.3

Authors: Thomas Limberg
Comments: 9 Pages. Language: German.

Like the title already tells, it is my goal to create a new foundation of mathematics. In this mathematical article we define the term "mathematical system" and introduce a simple mathematical system called System 1.1, in which a proof sketch of a proof of the existence of the universal set and the empty set within the system is made.
Category: Set Theory and Logic

[343] viXra:1903.0442 [pdf] submitted on 2019-03-24 06:19:51

Refutation of the Ordinal Turing Machine (Otm) on Set Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the sections on OTM-realizabilty and intuitionistic provability and axioms and systems of constructive set theories, we evaluate an inference rule and two propositions. None is tautologous. The refutes OTM on set theory in Hilbert space for intuitionistic logic. Therefore that approach produces non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[342] viXra:1903.0410 [pdf] submitted on 2019-03-22 09:58:01

Refutation of Coalgebraic Geometric Modal Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

Two definitions equations from Eqs. 4.4.1.2 and 4.4.4.2 as rendered are not tautologous, hence denying the monotone functor on KHaus. What follows is that the use of coalgebra to manufacture a geometric modal logic is refuted. Therefore the conjecture is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[341] viXra:1903.0406 [pdf] submitted on 2019-03-22 14:56:52

Refutation of Domain Theory

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate six equations for conjectures in five subsections of origins, bases of objects, axiomatic conditions, adjunctions, finite domains, and join-approximable relations. None is tautologous, hence refuting the domain theory of Dana Scott. Therefore, Scott's domain theory is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[340] viXra:1903.0335 [pdf] submitted on 2019-03-18 17:46:27

Refutation of Standard Induction, Coinduction and Mutual Induction, Coinduction

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the summary of standard and mutual induction and coinduction, we evaluated four formulas with non tautologous and hence refutations. Therefore these are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[339] viXra:1903.0302 [pdf] submitted on 2019-03-15 10:50:26

Refutation of Finitary, Non-Deterministic, Inductive Definitions of Ecst and Denial of CZF

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the elementary constructive set theory (ECST) of intuitionistic logic, we evaluate six axioms of equality for system CZF. None is tautologous. This refutes those axioms in set theory and by extension denies intuitionistic logic.
Category: Set Theory and Logic

[338] viXra:1903.0289 [pdf] submitted on 2019-03-14 07:25:10

Refutation of Coq Proof Assistant to Map Euclidean Geometry to Hilbert Space

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

For the relation of "a point is incident to a straight line", we find that proposition is not tautologous. This denies the conjectured approach of a constructive mapping Euclidean geometry into a Hilbert space and also refutes the Coq proof assistant as a bivalent tool. The conjecture and Coq are therefore non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[337] viXra:1903.0237 [pdf] submitted on 2019-03-12 15:08:14

Denial of Suzko's Problem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We examine a sentential logic description, as based on set theory, in support of Suzko's theorem that only two truth values are required as a universal logic. Under syntactic notions, we evaluate three definitions (monotonicity, transivity, permeability) out of six definitions (trivial are substitution-invariance, reflexivity, combined consequence relation). Monotonicity and transivity are not tautologous. Right-to-left permeability is not tautologous. What follows is that a Malinowski extension of mixed-consequence by relaxation of the two values for three logical values is spurious, especially due to the fact that Suzko's theorem is a conjecture based on the assumption of set theory. What also follows is that compositionality as based on Suzko-Scott reductions are not bivalent and exact, but rather a vector space and probabilistic. Our results point further to the equations analyzed as being non tautologous fragments of the universal logic VL4.
Category: Set Theory and Logic

[336] viXra:1903.0230 [pdf] submitted on 2019-03-12 23:23:19

Denial of Płonka Sums in Logics of Variable Inclusion and the Lattice of Consequence Relations

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the section on Płonka sums, we evaluate an equation derived thereform. It is not tautologous, hence coloring subsequent assertions in the conjecture. This means the non tautologous conjecture is a fragment of the the universal logic VŁ4.
Category: Set Theory and Logic

[335] viXra:1903.0229 [pdf] submitted on 2019-03-12 23:27:58

Denial that Modal Logics of Finite Direct Powers of ω Have the Finite Model Property

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the section partitions of frames, local finiteness, and the finite model property, we evaluate that definition. Because it is not tautologous, subsequent equations in the conjecture are denied. This means it is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[334] viXra:1903.0206 [pdf] submitted on 2019-03-11 21:04:51

Universal logic VŁ4

Authors: Colin James III
Comments: 20 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

This paper demonstrates why logic system VŁ4 is a universal logic composed of any refutation as a non-tautologous fragment. Recent advances are a definitive answer to criticism of logic Ł4, modal equations for lines and angles of the Square of Opposition, confirmation of the 24-syllogisms by updating Modus Cesare and Camestros, and proving that respective quantified and modal operators are equivalent. The parser Meth8 implements VŁ4 as the modal logic model checker Meth8/VŁ4. Over 430 assertions are tested in 2300 assertions with a refutation rate of 81%.
Category: Set Theory and Logic

[333] viXra:1903.0205 [pdf] submitted on 2019-03-11 21:36:16

Refutation of Induction Formulas in Elementary Arithmetic ea

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the introduction, we evaluate EA elementary arithmetic for induction formulas which are not tautologous. This further refutes the reflection property upon which subsequent assertions are based. These formulas constitute a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[332] viXra:1903.0146 [pdf] submitted on 2019-03-08 09:52:22

Refutation of Logical Theory Based on Compatible Consequence in Set Theory

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate canonically logical compatibility relations (CM) and complements (CRS), each in three sets of definitions. None is tautologous, so we avoid the subsequent ten relations. This refutes the "the possibility of a notion of compatibility that allows either for glutty or gappy reasoning". (By extension, paraconsistent logic is rendered untenable.) Therefore the bivalent standard notion of formal theory in logic is confirmed as allowing both assertion and denial as equally valid. In fact, this refutation further disallows injection of a bilateralist approach on many dimensions. This also indirectly reiterates that set theory is not bivalent, and hence derivations therefrom, such as the instant relations, are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[331] viXra:1903.0052 [pdf] submitted on 2019-03-03 10:15:49

Refutation of the Generating Positive Cone in Ordered Banach Space

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the background definitions in the ordered Banach space, we evaluate equations to produce the term named positive generating cone. It is not tautologous, hence refuting the model.
Category: Set Theory and Logic

[330] viXra:1903.0049 [pdf] submitted on 2019-03-03 15:04:19

Interval Sieve Algorithm - Creating a Countable Set of Real Numbers from a Closed Interval

Authors: Ron Ragusa
Comments: 5 Pages.

The Interval Sieve Algorithm is a method for generating a list of real numbers on any closed interval [ri, rj] where ri < rj, which can then be defined as the domain of the function f(x) = C. The purpose of this paper is to delineate the steps of the algorithm and show how it will generate a countable list from which the domain for the function f(x) = C can be defined. Having constructed the list we will prove that the list is complete, that it contains all the numbers in the interval [ri, rj].
Category: Set Theory and Logic

[329] viXra:1902.0482 [pdf] submitted on 2019-02-27 10:08:06

Refutation of the Power Set in Description Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate four, simple axioms of any Ω-model, including the operator Pow, to support the power set in description logic. None is tautologous, meaning the power set as asserted is refuted.
Category: Set Theory and Logic

[328] viXra:1902.0476 [pdf] submitted on 2019-02-27 12:29:45

Refutation of the Goldblatt-Thomason Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the introduction, we evaluate the terms forth and back as a duality. Neither is negation of the other, hence refuting the core basis of the Goldblatt-Thomason theorem. The question posed by it is further answered by the universal logic VŁ4 that expresses by modal axioms all first-order definable properties of a binary relation, due to equivalences of the respective quantifier and modal operator.
Category: Set Theory and Logic

[327] viXra:1902.0454 [pdf] submitted on 2019-02-26 14:40:56

Refutation of Ideals

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the definition of ideals. Two of three parts are not tautologous, refuting ideals.
Category: Set Theory and Logic

[326] viXra:1902.0451 [pdf] submitted on 2019-02-26 21:58:20

Refutation of Definable Operators on Stable Set Lattices

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the definitions for the modal operators on stable set lattices. The operators are not respective negations and hence refute the definitions.
Category: Set Theory and Logic

[325] viXra:1902.0450 [pdf] submitted on 2019-02-26 22:28:19

Confirmed Refutation of Kripke-Platek (KP) / Constructive Zermelo-Fraenkel Set Theory (CZF)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the set induction scheme for Kripke-Platek set theory (KP) and Constructive Zermelo-Fraenkel set theory (CZF). It is not tautologous. This confirms the previous refutation.
Category: Set Theory and Logic

[324] viXra:1902.0448 [pdf] submitted on 2019-02-26 23:58:12

On The Model Of Hyperrational Numbers With Selective Ultrafilter

Authors: Armen Grigoryants
Comments: 6 Pages.

In standard construction of hyperrational numbers using ultrapower we assume that the ultrafilter is selective. It makes possible to assign real value to any finite hyperrational number. So, we can consider hyperrational numbers with selective ultrafilter as extension of traditional real numbers. Also proved the existence of strictly monotonic or stationary representing sequence for any hyperrational number.
Category: Set Theory and Logic

[323] viXra:1902.0437 [pdf] submitted on 2019-02-25 09:12:52

Refutation of Smarandache Geometry

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the Smarandache algebra system without unit as basis of its geometry. On the right inverse operator in the dexter (right) digits, 1·1=1 contradicts 1·1=0. Hence Smarandache geometry is a probabilistic vector space and refuted as an exact, bivalent logic.
Category: Set Theory and Logic

[322] viXra:1902.0414 [pdf] submitted on 2019-02-24 15:48:39

Denial of Schematic Refutations of Formula Schemata

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a definition of the schematic formula of the proposed framework. It is not tautologous and hence denies these particular refutations of formula schema. The example of the pigeon hole principle, a trivial theorem, is also not refuted by the proposed framework.
Category: Set Theory and Logic

[321] viXra:1902.0402 [pdf] submitted on 2019-02-23 07:40:34

In Weighted First Order Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a formula of relativization as defined by structural induction which is not tautologous. Its use in weighted first order logic is refuted.
Category: Set Theory and Logic

[320] viXra:1902.0397 [pdf] submitted on 2019-02-23 17:11:22

Refutation of the Banach-Tarski Paradox

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the crucial claim of the proof in Step 3, as a fleshed out detail. It is not tautologous, nor is it contradictory. This means the claim is a non tautologous fragment of the universal logic VŁ4 and constitutes the briefest known refutation of the Banach-Tarski paradox.
Category: Set Theory and Logic

[319] viXra:1902.0379 [pdf] submitted on 2019-02-22 18:48:49

Refutation of a Complete Axiomatization of Reversible Kleene Lattices

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the main result theorem and two inequations as valid but not derivable from Kleene lattices. The theorem has an identical antecedent and consequent.. The two inequations are not tautologous and hence not valid, regardless of derivability without the union operator.
Category: Set Theory and Logic

[318] viXra:1902.0360 [pdf] submitted on 2019-02-21 18:30:13

Refutation: Constructive Zermelo-Fraenkel Set Theory (CZF) and Extended Church's Thesis (Ect)

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate constructive Zermelo-Fraenkel set theory (CZF) of intuitionistic logic. Of the eight CZF axioms, only the induction scheme is tautologous. From CZF axioms for infinity, set induction, and extensionality the deduction of one set denoted ω is not tautologous. An equation for extended Church's thesis (ECT) is not tautologous. This supports previous work that intuitionistic logic is not tautologous.
Category: Set Theory and Logic

[317] viXra:1902.0358 [pdf] submitted on 2019-02-21 19:00:40

Refutation of Provability of Consistency with Rosser's Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate provability of consistency with Rosser's theorem. A trivial theorem is found in the abstract, but a rule of necessitation and Rosser's theorem are not tautologous. While consistency is likely provable, the instant approach is refuted and thus not vindicating Hilbert.
Category: Set Theory and Logic

[316] viXra:1902.0281 [pdf] submitted on 2019-02-16 20:49:36

Refutation of Peirce's Abduction and Induction, and Confirmation of Deduction

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate definitions of C.S. Peirce for abduction, induction and deduction: all are inversions of the same sentence. However, when the connectives are changed to implication, abduction and induction are not tautologous, leaving deduction as the only form of tautologous inference in logic.
Category: Set Theory and Logic

[315] viXra:1902.0251 [pdf] submitted on 2019-02-14 13:19:23

Refutation of Bisimulation

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two definitions of bisimularity, both not tautologous. That refutes bisimulation, along with its proof tools of coinduction and Howe's congruence method.
Category: Set Theory and Logic

[314] viXra:1902.0249 [pdf] submitted on 2019-02-14 21:55:04

Meth8/VŁ4 on One and Three in Arithmetic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate arithmetic using 0 and 3 as binary 00 and 11.. Arithmetic holds in nine theorems. For division by zero, the result is Not(0 and 3).
Category: Set Theory and Logic

[313] viXra:1902.0243 [pdf] submitted on 2019-02-13 17:25:16

Refutation of Surveillance Objectives

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate an equivalence and theorem. Neither are tautologous. Hence the reduction of the multi-agent surveillance synthesis problem to solving single-sensor surveillance subgames is refuted. What follows is that surveillance objectives are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[312] viXra:1902.0227 [pdf] submitted on 2019-02-13 11:39:51

Refutation of Prevarieties and Quasivarieties of Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two papers by the same author group. For prevarieties, we test a theorem in M which is not tautologous. For quasivarieties, we test a definition and two theorems. The definition of a De Morgan monoid via an involution function is not tautologous. Theorems for the Dunn monoid and via Brouwerian (and Heyting) algebra are not tautologous. These results collectively refute prevarieties and quasivarieties in logic. What follows is that prevarieties and quasivarieties of logic are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[311] viXra:1902.0225 [pdf] submitted on 2019-02-12 16:21:24

Quantified Modifiers as Modal Operators† on Connectives in Modal Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

Quantified modifiers as modal operators †do not apply directly to connectives, but to sentences and variables in the general format of antecedent, connective, consequent. We present quantified expressions in that format by connective for two variables. The quantified expressions are equivalent to the quantifier as a modal modifier distributed on the variables.
Category: Set Theory and Logic

[310] viXra:1902.0208 [pdf] submitted on 2019-02-12 11:45:36

Refutation of the Unification Type of Simple Symmetric Modal Logics

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the definitions of two new modal connectives as box-plus and box-minus. Neither is tautologous and both are equivalent. This refutes the unification type of simple symmetric modal logics and implies it is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[309] viXra:1902.0203 [pdf] submitted on 2019-02-11 21:51:15

Refutation of Relevance Logic R and Models

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a definition and model formula for relevance logic R which are not tautologous. What follows is that logic R is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[308] viXra:1902.0197 [pdf] submitted on 2019-02-11 08:12:37

Refutation of the Method of Forcing

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two examples of paradoxes to be resolved by the method of forcing. The first is not tautologous, and the second is a contradiction so not proved as a paradox. This means the forcing method is refuted because it cannot coerce the two examples into abstract proofs and with the ultimate goal to produce larger truth values. What follows is that the forcing method is better suited for paraconsistent logics which are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[307] viXra:1902.0196 [pdf] submitted on 2019-02-11 08:33:38

Refutation of Subset Models for Justification Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two axioms for justification logic which are not tautologous. This means subset models for justification logic are refuted.
Category: Set Theory and Logic

[306] viXra:1902.0177 [pdf] submitted on 2019-02-10 12:04:33

Refutation of Reverse Mathematics on Measurability Theory and Computability Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the authors' definition of reverse mathematics in their anticipation of applying it to measurability and computability theory. The argument taking two equations to define reverse mathematics is not tautologous. Therefore to apply it to measure and computability theory is meaningless.
Category: Set Theory and Logic

[305] viXra:1902.0167 [pdf] submitted on 2019-02-09 06:26:40

Denial of the Refutation of Coherence in Modal Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the refutation of coherence in modal logic as based on weakly transitive logics using a ternary term to admit finite chains. The term is not tautologous, thereby denying the refutation. What follows is that K, KT, K4, and S4 (and S5) are tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[304] viXra:1902.0141 [pdf] submitted on 2019-02-07 18:52:35

Denial of Logic System PŁ4

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the logic system PŁ4 as refuting and replacing VŁ4. Seven modal theses and four axioms are not tautologous and contrary to those of PŁ4. This denies that PŁ4 refutes VŁ4, refutes PŁ4, and justifies VŁ4 as containing the non bivalent fragment named PŁ4.
Category: Set Theory and Logic

[303] viXra:1902.0140 [pdf] submitted on 2019-02-07 21:54:41

Two VŁ4 Theorems not in S5

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two equations not found as theorems in S5; both are theorems in VŁ4.
Category: Set Theory and Logic

[302] viXra:1902.0126 [pdf] submitted on 2019-02-07 09:15:50

Refutation of Disturbance as a Feature

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate four binary equations as not tautologous. The refutes the formal description of verifiability for disturbance as a feature.
Category: Set Theory and Logic

[301] viXra:1902.0119 [pdf] submitted on 2019-02-06 18:13:16

Refutation of the Axiomatic Theory of Betweenness

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate sixteen equations as axioms and conclusions, for nine as not tautologous. This refutes the axiomatic theory of betweenness.
Category: Set Theory and Logic

[300] viXra:1902.0096 [pdf] submitted on 2019-02-05 20:48:51

Refutation of the Axiom of Extensionality

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the axiom of extensionality. The equation we tested is not tautologous. This refutes the axiom of extensionality.
Category: Set Theory and Logic

[299] viXra:1902.0087 [pdf] submitted on 2019-02-05 05:57:27

Confirmation of the Failure of Modus Ponens When the Consequent is Itself a Conditional Sentence

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We confirm the failure of modus ponens when the consequent is itself a conditional sentence. The reason a repeated consequent does not produce tautology is because it dilutes the original sentence to assume incorrectly other plausible consequents.
Category: Set Theory and Logic

[298] viXra:1902.0082 [pdf] submitted on 2019-02-05 12:01:02

Refutation of the Parameter Free ZFC0

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the parameter free ZFC0. We test the parameter free schema of comprehension and of replacement. Neither are tautologous. This refutes ZFC0.
Category: Set Theory and Logic

[297] viXra:1902.0080 [pdf] submitted on 2019-02-04 21:26:11

Refutation of Correctness from Pratt-Floyd-Hoare Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the Pratt-Floyd-Hoare logic aimed at correctness of computer programs. For Pratt, four equations are tested, and for Hoare one is tested. None are tautologous. This refutes the Pratt-Floyd-Hoare logic for correctness.
Category: Set Theory and Logic

[296] viXra:1902.0066 [pdf] submitted on 2019-02-04 08:28:51

Refutation of Dynamic Modal Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two definitions of dynamic modal logic on which axioms are built. The definitions are not tautologous and moreover logically equivalent. This refutes dynamic modal logic.
Category: Set Theory and Logic

[295] viXra:1902.0065 [pdf] submitted on 2019-02-04 08:50:09

Construction of Sets and Axiomatic System

Authors: Baekhyun Lee
Comments: 15 Pages.

This paper is written to show that concept of sets can be constructed with a few axioms, and also the axioms can be treated as a set.
Category: Set Theory and Logic

[294] viXra:1902.0037 [pdf] submitted on 2019-02-02 22:19:59

Refutation of Confluence in Rewrite Systems

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate confluence in two definitions and one theorem, none which is tautologous. This refutes the approach of confluence as a central property of rewrite systems.
Category: Set Theory and Logic

[293] viXra:1902.0022 [pdf] submitted on 2019-02-01 18:44:04

Refutation of Mapping Definable Functions as Neighbourhood Functions

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two applied equations of stopping and closure functions of bar recursion. None is tautologous. This refutes the approach of mapping functions as such.
Category: Set Theory and Logic

[292] viXra:1902.0012 [pdf] submitted on 2019-02-01 07:53:15

Mathematical Induction as a Higher-Order Logical Principle Based on Permutations of F>F=T.

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

The principle of induction is tautologous as a permutation by way of the generic format of F>F=T.
Category: Set Theory and Logic

[291] viXra:1902.0006 [pdf] submitted on 2019-02-01 10:49:35

Refutation of Recursive Comprehension in Second-Order Arithmetic

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate 11 basic first-order axioms of which nine are not tautologous. Recursive comprehension, as an abstraction of mathematical induction, is derived therefrom in second-order arithmetic and is not tautologous. This refutes the use of recursive comprehension in second-order arithmetic. Reverse mathematics relies on recursive comprehension and hence is also refuted.
Category: Set Theory and Logic

[290] viXra:1901.0467 [pdf] submitted on 2019-01-31 09:57:54

Refutation of Spatial Relations and Claims in Distributive Mereotopology

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the contact algebra logic of RCC-8 for tangential and non-tangential spatial relations. The respective representations in T0 spaces for four equations to be equivalent. Two only are equivalent, with the iff implication chain as not tautologous. These results refute distributive mereotopology.
Category: Set Theory and Logic

[289] viXra:1901.0460 [pdf] submitted on 2019-01-30 18:24:18

Refutation of De Finettian Logics of Indicative Conditionals

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate De Finettian logics which specifies four conditionals, as attributed to Aristotle, Boethius, Cooper-Cantwell, Jeffrey, and six axioms. None is tautologous. This refutes their use to justify connexive logic.
Category: Set Theory and Logic

[288] viXra:1901.0458 [pdf] submitted on 2019-01-30 20:30:32

Refutation of the Quantifier Most

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two equivalent semantics for "Most A are B" which while logically equivalent are neither tautologous. Hence the quantifier Most is refuted.
Category: Set Theory and Logic

[287] viXra:1901.0446 [pdf] submitted on 2019-01-29 20:07:52

Refutation of the Interval for Model Checking

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate three sub-interval relations named reflexive, proper or irreflexive, and strict. None is tautologous. This refutes those relations and model checking therefrom.
Category: Set Theory and Logic

[286] viXra:1901.0442 [pdf] submitted on 2019-01-29 07:21:53

Refutation of Sabotage Modal Logic (Revisited)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate six equations for a definition and properties of sabotage modal logic. The special symbols ■ and ♦ act as functions, so we assign them variable names. Because none is tautologous, sabotage modal logic is refuted.
Category: Set Theory and Logic

[285] viXra:1901.0440 [pdf] submitted on 2019-01-29 08:02:21

Refutation of Poison Modal Logic (PML)

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate four formulas as validities of poison modal logic (PML). None is tautologous, thereby refuting poison modal logic (PML).
Category: Set Theory and Logic

[284] viXra:1901.0434 [pdf] submitted on 2019-01-28 18:43:38

Refutation of Approximations of Theories

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the definition of T-approximations which is not tautologous, thereby refuting the approximations of theories.
Category: Set Theory and Logic

[283] viXra:1901.0423 [pdf] submitted on 2019-01-28 08:40:52

Refutation of an Epistemic Logic for Knowledge and Probability

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate axioms and definitions of an epistemic logic for knowledge and probability. Two equations are not tautologous, hence refuting the proposed epistemic logic.
Category: Set Theory and Logic

[282] viXra:1901.0382 [pdf] submitted on 2019-01-26 12:07:00

Refutation of Dempster-Shafer Belief and Plausibility Theory

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate Dempster-Shafer belief and plausibility functions. Three definitions are not tautologous. This refutes Dempster-Shafer belief and plausibility theory.
Category: Set Theory and Logic

[281] viXra:1901.0379 [pdf] submitted on 2019-01-25 21:34:32

Refutation of a Modal Logic System for Reasoning About the Degree of Blameworthiness

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a modal logic system for the blameable coalition of an outcome if there is a strategy to prevent it and where the degree of blameworthiness is measured by costs of prevention or sacrifice. Of eight axioms, three are not tautologous, hence refuting the approach. We do not consider the claimed technical result of a completeness theorem.
Category: Set Theory and Logic

[280] viXra:1901.0342 [pdf] submitted on 2019-01-23 11:57:15

Refutation of Descriptive Unions in Descriptively Near Sets

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate an intersection operator named descriptive union for descriptively near sets. A proof of seven properties contains two trivial tautologies and the rest as not tautologous. The refutes the descriptive intersection operator and descriptively near sets on which it is based. This also casts doubt on the derived slug of math and physics papers as spawned at archiv, researchgate, and vixra.
Category: Set Theory and Logic

[279] viXra:1901.0308 [pdf] submitted on 2019-01-21 10:35:28

Refutation of Block Argumentation

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the approach of block argumentation using the legal example of a popular case. Three scenarios are not tautologous. We attempt to resuscitate the method by substitution of generic block argumentation and also by testing the consequent parts separately. The equations were not tautologous, hence refuting the block argumentation approach as presented.
Category: Set Theory and Logic

[278] viXra:1901.0192 [pdf] submitted on 2019-01-13 15:15:58

Refutation of Hexagons of Opposition for Statistical Modalities

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

Abstract: We evaluate pragmatic hypotheses in the evolution of science as based on probabilistic squares and hexagons of opposition under coherence. Neither conjecture is tautologous, and hence both are refuted.
Category: Set Theory and Logic

[277] viXra:1901.0190 [pdf] submitted on 2019-01-13 20:27:25

An Even Number Greater Than 2 Equals the Sum of Two Prime Numbers.

Authors: Wu Ye TangYin
Comments: 16 Pages.

On the contrary, if the natural number 2N is not equal to the prime number, On the contrary, prime numbers can only appear in odd numbers.
Category: Set Theory and Logic

[276] viXra:1901.0132 [pdf] submitted on 2019-01-11 05:33:24

Refutation of a Modal Aleatoric Calculus for Probabilistic Reasoning: Extended Version

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We instruct troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a modal aleatoric calculus for probabilistic reasoning using the assumption of probabilistic definitions as P(¬α) = 1−P(α). Five equations in Lemma 1 and its argument are tested. All equations are not tautologous, hence refuting the calculus.
Category: Set Theory and Logic

[275] viXra:1901.0092 [pdf] submitted on 2019-01-07 11:41:57

Denial of the Alleged Łukasiewicz Nightmare in System Ł4

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We instruct troll Mikko at Disqus to read the article four times before hormonal typing.)

The alleged Łukasiewicz nightmare of (◇p&◇q)→◇(p&q) is not tautologous in Prover9; however, the equation recast in one variable as (◇p&◇~p)→◇(p&~p) is tautologous. In Meth8/VŁ4, both propositions are tautologous. This speaks for Meth8/VŁ4, based on the corrected modern Square of Opposition as an exact bivalent system, as opposed to Prover9, based on the uncorrected modern Square of Opposition as an inexact probabilistic vector space.
Category: Set Theory and Logic

[274] viXra:1901.0069 [pdf] submitted on 2019-01-05 10:13:15

Monoidal Structures of Logic

Authors: Antoine Balan
Comments: 2 pages, written in french

The tables of truth are relations of a monoid M.
Category: Set Theory and Logic

[273] viXra:1901.0020 [pdf] submitted on 2019-01-02 08:43:08

Relational Structures

Authors: J.A.J. van Leunen
Comments: 3 Pages. This is part of the Hilbert Book Model Project

Lattices form the foundation of important theories. One is classical logic. Another is quantum logic, which is an orthomodular lattice.
Category: Set Theory and Logic

[272] viXra:1812.0440 [pdf] submitted on 2018-12-27 14:03:53

Refutation of a Minimal Non-Contingency Logic System

Authors: Colin James III
Comments: 1 Page. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate a minimal non-contingency logic system based on its unique definition which is not tautologous and hence reject it.
Category: Set Theory and Logic

[271] viXra:1812.0429 [pdf] submitted on 2018-12-26 21:59:10

The New Bivalent, Three-Valued Logic VŁ3

Authors: Colin James III
Comments: 3 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We build and test a new bivalent three-valued logic named VŁ3. Recent advances are support of the classical tautologies, modal definitions, the law of excluded fourth, and extended contradiction principle.
Category: Set Theory and Logic

[270] viXra:1812.0423 [pdf] submitted on 2018-12-25 17:37:59

The Curvature and Dimension of 2-Surfaces

Authors: Shawn Halayka
Comments: 2 Pages.

The curvature of 2-surfaces can lead to fractional dimension.
Category: Set Theory and Logic

[269] viXra:1812.0371 [pdf] submitted on 2018-12-20 07:18:22

Refutation of Lambda λ-Calculus and Lisp

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate McCarthy's lambda λ-calculus in six equations. Five of the equations are not tautologous, and one equation is a trivial tautology. McCarthy's three-valued logic is not bivalent on which the LISP programming language is implemented, and hence also flawed.
Category: Set Theory and Logic

[268] viXra:1812.0314 [pdf] submitted on 2018-12-18 19:54:48

Rejection of Trivial Objections to Modal Logic Ł4

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate objections to the modal logic Ł4 by six equations in contra arguments which we reject as not tautologous. The concluding equation invoked as (((p=p)=(q=q)))=(r=r))=((p=q)=r) is not tautologous. We reject the trivial conclusion that "modal syllogisms with both necessary premises and with mixed premises cannot be distinguished while one is necessary and another assertoric[:] Łukasiewicz’ modal logic is useless for investigating Aristotelian modal syllogistic". Hence we use our VŁ4 to invalidate objections to itself.
Category: Set Theory and Logic

[267] viXra:1812.0275 [pdf] submitted on 2018-12-17 00:43:28

Confirmation of the Łukasiewicz Square of Opposition Via Logic VŁ4

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the existential import of the revised, modern square of opposition. We confirm that the Łukasiewicz syllogistic was intended to apply to all terms. What follows is that Aristotle was mistaken in his mapping of vertices, which we correct and show fidelity to Aristotle's intentions.
Category: Set Theory and Logic

[266] viXra:1812.0267 [pdf] submitted on 2018-12-15 08:57:19

Refutation of Pure Alethic Modal Logic (Pam)

Authors: Colin James III
Comments: 1 Page. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the formula □p → p → ◊p as the backbone of pure alethic modal logic (PAM). Two inconsistent results arise from different orders of precedence: a result of not tautologous, not contradictory; and a tautologous result. That ambiguity refutes PAM.
Category: Set Theory and Logic

[265] viXra:1812.0217 [pdf] submitted on 2018-12-12 13:52:02

Refutation of Graded Modal Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate definitions of five frame classes and a main theorem of satisfiability. Three of the five frame classes are not tautologous and the designated example for satisfiability is not tautologous. This refutes graded modal logic.
Category: Set Theory and Logic

[264] viXra:1812.0214 [pdf] submitted on 2018-12-12 21:38:39

Refutation of Lyndon Interpolation

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the Lyndon interpolation on the logic GL. Each is not tautologous, and the combination is not tautologous, hence rendering both refuted.
Category: Set Theory and Logic

[263] viXra:1812.0209 [pdf] submitted on 2018-12-11 12:15:20

Refutation of the Hahn-Banach Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the Hahn-Banach theorem. Without or with the universal quantifiers, the equations are not tautologous. This refutes the Hahn-Banach theorem.
Category: Set Theory and Logic

[262] viXra:1812.0205 [pdf] submitted on 2018-12-11 22:27:00

Refutation of Agm Postulates and Levi and Harper Bridging Principles

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the AGM logic system in eight postulates and two bridging principles. The postulates named success, inclusion, vacuity, and inconsistency and the Levi and Harper bridging principles are not tautologous, hence refuting the AGM logic system.
Category: Set Theory and Logic

[261] viXra:1812.0204 [pdf] submitted on 2018-12-11 22:28:46

Resolution of Moore's Paradox as a Theorem

Authors: Colin James III
Comments: 1 Page. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate Moore's paradox, as touted by Wittgenstein, with Hintikka's omissive or commissive logical forms as "P and NOT(belief in P)" or "P and belief in NOT-P". The former as antecedent (contradictory) and the latter as consequent (neither contradictory nor tautologous) imply tautology, a theorem.
Category: Set Theory and Logic

[260] viXra:1812.0168 [pdf] submitted on 2018-12-09 10:12:03

Affirmation of the Craig Interpolation Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the Craig interpolation theorem, find a mistake in a Craig lemma as rendered by Feferman, and affirm the theorem.
Category: Set Theory and Logic

[259] viXra:1812.0165 [pdf] submitted on 2018-12-09 16:20:16

Refutation of the modal logic GL2

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the modal logic GL2 in two axioms and for satisfiability. One of the two axioms is not tautologous, and the five formulas for PSpace complete satisfiability are not tautologous. Hence GL2 is refuted.
Category: Set Theory and Logic

[258] viXra:1812.0161 [pdf] submitted on 2018-12-10 04:29:36

An Application for Medical Decision Making with the Fuzzy Soft Sets

Authors: Murat Kirisci
Comments: 6 Pages.

In the present study, for the medical decision making problem, the proposed technique related to the fuzzy soft set by Celik-Yamak through Sanchez’s method was used. The real dataset which is called Cleveland heart disease dataset applied in this problem.
Category: Set Theory and Logic

[257] viXra:1812.0152 [pdf] submitted on 2018-12-08 20:08:04

Refutation of the Definition of Sacchetti's Modal Logics of Provability

Authors: Colin James III
Comments: 1 Page. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the definition of Sacchetti's modal logics of provability. It is not tautologous. Therefore it is a mistake to use it as a basis for constructing fixed point procedures.
Category: Set Theory and Logic

[256] viXra:1812.0047 [pdf] submitted on 2018-12-03 19:43:40

Refutation of the Lonely Runner Conjecture with Three Runners

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the conjecture of the lonely runner with three runners. We do not assume a runner may be stationary as a no-go contestant. The result is that the conjecture diverges from tautology by one logical value and hence is refuted. We also assume a runner can be stationary; those non tautologous results also refute the conjecture.
Category: Set Theory and Logic

[255] viXra:1812.0038 [pdf] submitted on 2018-12-02 08:48:13

Refutation of a Modal Logic for Partial Awareness from Published Example

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate a modal logic for partial awareness from a published example. The definitions and conjectures are not tautologous. We show how to exclude a priori logical clauses to promote a perhaps unintended tautology for the example. However, our evaluation does not rely on modal operators, suggesting that the system as proffered should be renamed to a logic for awareness, without the word modal.
Category: Set Theory and Logic

[254] viXra:1812.0016 [pdf] submitted on 2018-12-01 18:50:59

Refutation of the Frauchiger-Renner Thought Paradox as a Quantum Model

Authors: Colin James III
Comments: 4 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We use modal logic to evaluate a quantum model of the Frauchiger-Renner thought experiment as not a contradiction (paradox) and not a tautology (theorem). The example misapplies the Born rule which we refute elsewhere.
Category: Set Theory and Logic

[253] viXra:1812.0007 [pdf] submitted on 2018-12-01 03:10:43

Refutation of Modal Forms on Vietoris Space

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We use modal logic to evaluate the defined modal forms on Vietoris space to find them not tautologous.
Category: Set Theory and Logic

[252] viXra:1811.0489 [pdf] submitted on 2018-11-28 09:23:10

"Refutation of Intuitionistic Logic on a "Transfinite Argument""

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate intuitionistic logic via Hilbert's "transfinite argument" and Komogorov's implementation. None of the axioms is tautologous.
Category: Set Theory and Logic

[251] viXra:1811.0442 [pdf] submitted on 2018-11-27 19:36:39

Refutation of the Two-Sided Page Paradox

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the two-sided page conjecture that: if either the front page implies the back page is false or the back page implies the front page is true is a paradox (contradiction). The conjecture is a theorem and hence refuted as a paradox.
Category: Set Theory and Logic

[250] viXra:1811.0439 [pdf] submitted on 2018-11-27 22:44:02

Remark on Five Applications of Neutrosophic Logic: in Cultural Psychology, Economics Theorizing, Conflict Resolution, Philosophy of Science, and Cosmology

Authors: Victor Christianto, Florentin Smarandache
Comments: 10 Pages. This paper was written under guidance of Holy Spirit

In this short article, we review five applications of NFL which we have explored in a number of papers. Hopefully the readers will find a continuing line of thoughts in our research in the last few years.
Category: Set Theory and Logic

[249] viXra:1811.0382 [pdf] submitted on 2018-11-23 05:59:45

Refutation of Two Modern Modal Logics: "JYB4" and the Follow-on "AR4"

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the four-valued logic systems of J.-Y. Béziau and F. Schang as JYB4 and AR4. JYB4 named the four-valued modal logic Ł4 as Łukasiewicz's nightmare because of the alleged absurdity of (◇p&◇q) → ◇(p&q). A model checking system is then framed based on 0± and 1±. We show (◇p&◇q) → ◇(p&q) is equivalent to (◇p&◇~p) → ◇(p&~p) with (◇p&◇q) = ◇(p&q) as a theorem. AR4 was a doxastic logic follow-on to JYB4. We name these modern modal logic systems as Béziau's nightmare and Schang's nightmare.
Category: Set Theory and Logic

[248] viXra:1811.0374 [pdf] submitted on 2018-11-23 13:43:15

Rule of Necessitation: a Non-Contingent Truthity, But not a Tautology

Authors: Colin James III
Comments: 4 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

1. The axiom or rule of necessitation N states that if p is a theorem, then necessarily p is a theorem: If ⊢ p then ⊢ ◻p. We show this is non-contingent (a truthity), but not tautologous (a proof). We evaluate axioms (in bold) of N, K, T, 4, B, D, 5 to derive systems (in italics) of K, M, T, S4, S5, D. We conclude that N the axiom or rule of necessitation is not tautologous. Because system M as derived and rendered is not tautologous, system G-M also not tautologous. What follows is that systems derived from using M are tainted, regardless of the tautological status of the result so masking the defect, such as systems S4, B, and S5. We also find that Gentzen-sequent proof is suspicious, perhaps due to its non bi-valent lattice basis in a vector space.
Category: Set Theory and Logic

[247] viXra:1811.0269 [pdf] submitted on 2018-11-17 13:14:25

Refutation of Superposition as Glue in Matita Theorem Prover

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the substitution lemma for the successor function, smart application of inductive hypotheses, and proof traces of a complex example in the Matita standard library. Results are not tautologous, hence refuting superposition.
Category: Set Theory and Logic

[246] viXra:1811.0264 [pdf] submitted on 2018-11-17 18:17:31

Refutation of Metamath Theorem Prover

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate six conjectures and one theorem, as proffered by Metamath staff. The conjectures are not tautologous. The Tarski-Grothendieck theorem is also not tautologous. Metamath fails our analysis.
Category: Set Theory and Logic

[245] viXra:1811.0220 [pdf] submitted on 2018-11-14 18:40:03

Shorter Refutation of the Löb Theorem and Gödel Incompleteness by Substitution of Contradiction

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Löb’s theorem □(□X→X)→□X and Gödel’s incompleteness as □(□⊥ →⊥)→□⊥ are refuted.
Category: Set Theory and Logic

[244] viXra:1811.0154 [pdf] submitted on 2018-11-09 08:30:47

Find the extra shape (In Russian)

Authors: V. A. Kasimov
Comments: 2 Pages. in Russian

First, the task will put you in a dead end. No figure clearly stands out from the total number more than others. It's not as easy as it may seem in the first seconds.
Category: Set Theory and Logic

[243] viXra:1811.0078 [pdf] submitted on 2018-11-05 15:44:40

Refutation of Behavioral Mereology

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

If P≤P′ and Q′≤Q, proposition <>P'P<>QP' = <>QP is equivalent to []P'P[]QP' = []QP and respectively not tautologies.
Category: Set Theory and Logic

[242] viXra:1811.0075 [pdf] submitted on 2018-11-05 20:42:06

Refutation of the Blok-Esakia Theorem for Universal Classes

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Grzegorczyk (gzr) algebras as used for support and the Blok-Esakia theorems are not confirmed as tautologies and hence refuted.
Category: Set Theory and Logic

[241] viXra:1811.0059 [pdf] submitted on 2018-11-04 19:11:29

Refutation of First-Order Continuous Induction on Real Closed Fields

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

By mapping definitions, theorems, and propositions into Meth8/VŁ4, we refute the first-order continuous induction principle on real closed fields.
Category: Set Theory and Logic

[240] viXra:1811.0020 [pdf] submitted on 2018-11-01 10:35:55

New Axioms in the Set Theory

Authors: Antoine Balan
Comments: 2 pages, written in english

We propose three new axioms in set theory, axiomatising the measure theory of the Hilbert spaces.
Category: Set Theory and Logic

[239] viXra:1810.0350 [pdf] submitted on 2018-10-21 12:28:45

First Cryptocurrency Project Based on Strict Mathematical Rules

Authors: Linda Wiklund
Comments: 9 Pages.

First Cryptocurrency project based on strict mathematical rules
Category: Set Theory and Logic

[238] viXra:1810.0328 [pdf] submitted on 2018-10-20 21:05:52

Refutation of Set Theory by Supremum and Infimum

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Set theory is refuted by the supremum and infimum artifacts.
Category: Set Theory and Logic

[237] viXra:1810.0309 [pdf] submitted on 2018-10-19 12:19:11

Refutation of the Second Incompleteness Theorem by Gödel Logic

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Gödel's second incompleteness theorem as based on the minimal modal logic to express the Löb axiom is not tautologous. Subsequent substitutions into the Löb axiom along with Hájek's earlier lemma raise further suspicion about Gödel-justification logic.
Category: Set Theory and Logic

[236] viXra:1810.0082 [pdf] submitted on 2018-10-07 04:14:05

Refutation of the Sliding Scale Theorem in Law

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

The sliding scale theorem, and as implemented in fuzzy logic, is refuted.
Category: Set Theory and Logic

[235] viXra:1810.0072 [pdf] submitted on 2018-10-05 19:34:31

Refutation of Existentially Closed De Morgan Algebras

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Existentially closed De Morgan algebras are refuted.
Category: Set Theory and Logic

[234] viXra:1810.0067 [pdf] submitted on 2018-10-06 02:40:03

Refutation of Short Circuit Evaluation for Propositional Logic by Commutative Variants

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

The short circuit evaluation for propositional logic by commutative variants is not tautologous, and thereby refuted.
Category: Set Theory and Logic

[233] viXra:1809.0263 [pdf] submitted on 2018-09-12 06:11:47

Refutation of Mapping mu-Calculus Onto Second-Order Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

When mapping mu-calculus onto second-order logic, we show use of the fixpoint operator as untenable. What follows is the effective refutation of mapping mu-calculus onto second-order logic.
Category: Set Theory and Logic

[232] viXra:1809.0257 [pdf] submitted on 2018-09-12 11:52:54

Refutation of Completeness for Non-Deterministic Logic

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We show that the four-valued, non-deterministic semantics for modal logic are not complete. The demonstration uses contradictions based on Carnielli's paraconsistent logic. What follows is that infinite non-deterministic matrices are by definition incomplete.
Category: Set Theory and Logic

[231] viXra:1808.0090 [pdf] submitted on 2018-08-07 15:06:59

Refutation of Lean Theorem Prover from Microsoft

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Respond to this author's email address: info@ersatz-systems dot com. (We instruct troll Mikko at Disqus to read the entire article twice before she starts typing.)

These examples were found not tautologous: (∀x,px→r)↔(∃x,px)→r; (∃x,px→r)↔(∀x,px)→r; (∃x,r→px)↔(r→∃x,px). Hence Lean prover from Microsoft is not bivalent and refuted.
Category: Set Theory and Logic

[230] viXra:1808.0047 [pdf] submitted on 2018-08-03 22:08:52

Shortest Refutations of the Zermelo-Fraenkel (ZF) Axioms

Authors: Colin James III
Comments: 4 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The nine Zermelo-Fraenkel (ZF) axioms evaluated are not tautologous.
Category: Set Theory and Logic

[229] viXra:1807.0514 [pdf] submitted on 2018-07-30 18:56:24

Confirmation that Every Straight Line Through a Point Inside a Circle Intersects the Circumference

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

"An interior point inside the circle and an exterior point outside the circle imply that every straight line intersecting both points intersects a point on the circle." This conjecture is comfirmed.
Category: Set Theory and Logic

[228] viXra:1807.0509 [pdf] submitted on 2018-07-29 06:56:29

Confirmation of Playfair's Axiom

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

Per "De Morgan that this proposition is logically equivalent to Playfair’s axiom. ... Let X be the set of pairs of distinct lines which meet and Y the set of distinct pairs of lines each of which is parallel to a single common line. If z represents a pair of distinct lines, then the statement, For all z, if z is in X then z is not in Y, is Playfair's axiom, and its logically equivalent contrapositive, For all z, if z is in Y then z is not in X, is Euclid I.30, the transitivity of parallelism." This is confirmed as tautologous.
Category: Set Theory and Logic

[227] viXra:1807.0499 [pdf] submitted on 2018-07-29 12:12:31

Confirmation of the Triangle Inequality

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

"[T]he triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side." We rewrite this as contradiction F implies tautology T using absolute values of distances: "If qr is not greater than pq with pr, then both qr is greater than pq and qr is greater than pr." Hence the triangle inequality is confirmed as tautologous. Remark: This exercise indirectly speaks to the fact that the vector space is not bivalent.
Category: Set Theory and Logic

[226] viXra:1807.0478 [pdf] submitted on 2018-07-28 16:50:58

Refutation of the Heisenberg Principle as a no-go Axiom, and Its Trivial Replacement Theorem

Authors: Colin James III
Comments: 2 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The Heisenberg uncertainty principle "states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa". This is refuted as not tautologous. In terms of the statistic of deviation, a trivial replacement theorem is: "If position is greater than momentum, then position and momentum are greater than position or momentum."
Category: Set Theory and Logic

[225] viXra:1807.0472 [pdf] submitted on 2018-07-27 15:42:55

Refutation of Relevance Logic Via Routely and Meyer

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The conjecture "A → B is true at a world a if and only if for all worlds b and c such that Rabc (R is the accessibility relation) either A is false at b or B is true at c" is not tautologous.
Category: Set Theory and Logic

[224] viXra:1807.0452 [pdf] submitted on 2018-07-26 23:41:32

Implication Combinations Derived from (P>q)>r

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

12 tautologous equations are named the general forms of (p>q)>r by implication on Meth8/VŁ4. (They are not listed here because the abstract block mungs line spacing.)
Category: Set Theory and Logic

[223] viXra:1807.0431 [pdf] submitted on 2018-07-25 18:48:36

Trivial Proofs

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

For all p, p+p=p (1.1). #p>((p+p)=p) ; TTTT TTTT TTTT TTTT (1.2). For one p, p+p=p (2.1). %p>((p+p)=p) ; TTTT TTTT TTTT TTTT (2.2). Axiom of associativity (3.1). ((p+q)+r)=(p+(q+r)) ; TTTT TTTT TTTT TTTT (3.2).
Category: Set Theory and Logic

[222] viXra:1807.0418 [pdf] submitted on 2018-07-24 20:38:58

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. Source code is provided.
Category: Set Theory and Logic

[221] viXra:1807.0417 [pdf] submitted on 2018-07-24 22:55:24

A Dual Identity Based Symbolic Understanding of the Gödel’s Incompleteness Theorems, P-NP Problem, Zeno’s Paradox and Continuum Hypothesis

Authors: Arun Uday
Comments: 29 Pages.

A semantic analysis of formal systems is undertaken, wherein the duality of their symbolic definition based on the “State of Doing” and “State of Being” is brought out. We demonstrate that when these states are defined in a way that opposes each other, it leads to contradictions. This results in the incompleteness of formal systems as captured in the Gödel’s theorems. We then proceed to resolve the P-NP problem, which we show to be a manifestation of Gödel’s theorem itself. We then discuss the Zeno’s paradox and relate it to the same aforementioned duality, but as pertaining to discrete and continuous spaces. We prove an important theorem regarding representations of irrational numbers in continuous space. We extend the result to touch upon the Continuum Hypothesis and present a new symbolic conceptualization of space, which can address both discrete and continuous requirements. We term this new mathematical framework as “hybrid space”.
Category: Set Theory and Logic

[220] viXra:1807.0410 [pdf] submitted on 2018-07-23 22:31:40

Refutation of Stit Logic (Sees to it That)

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

From four papers in the relevant literature for 2017-2018, 13 equations are not tautologous. For example, an equation claimed for necessitation as p>[]p is TNTN TNTN TNTN TNTN (not all T proof values). Hence Stit logic is refuted
Category: Set Theory and Logic

[219] viXra:1807.0392 [pdf] submitted on 2018-07-24 11:54:04

Refutation of Levi-Identity and Agm Postulates

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The Levi-identity as rendered is not tautologous, as a basis for the subsequent AGM expressions. Four of the eight AGM equations are not tautologous, hence refuting them as a postulate system for fictional logic.
Category: Set Theory and Logic

[218] viXra:1807.0380 [pdf] submitted on 2018-07-22 08:28:58

Refutation of Connexive Logic Based on Wansing's Nightmare

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Connexive logic turns on Wansing's nightmare of ∼(A→B)↔(A→∼B), or the falsity-weakend ∼(A→B)↔(∼A→B), both of which are not tautologous.
Category: Set Theory and Logic

[217] viXra:1807.0373 [pdf] submitted on 2018-07-22 17:06:11

Refutation of Aristotle's and Boethius' Theses

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com

Aristotle's Theses and Boethius' Theses are below. ~(~A> A)=(A=A) ; TNCF TNCF TNCF TNCF (AT) . ~( A>~A)=(A=A) ; FCNT FCNT FCNT FCNT (AT)'. (A> B)>~(A>~B) ; FCNT FCNT FCNT FCNT (BT) . (A>~B)>~(A> B) ; FCNT FCNT FCNT FCNT (BT)'. Eqs. AT, AT', BT, and BT' are not tautologous, hence refuting those theses of Aristotle and Boethius. Because ~AT = AT' = BT = BT', what follows is that using conditionals to justify connexive logic makes Wansing's nightmare worse.
Category: Set Theory and Logic

[216] viXra:1807.0365 [pdf] submitted on 2018-07-21 09:51:40

Refutation of the Improved Adams Hypothesis of Conditional Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The seven equations as rendered are not tautologous. This means the probability axioms of Popper and McGee and the improved hypothesis of Adams are refuted.
Category: Set Theory and Logic

[215] viXra:1807.0357 [pdf] submitted on 2018-07-21 23:55:29

Refutation of Conditional Events in Quantum Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The following conjectures are refuted: Definition of simultaneous verifiability; Definition of simultaneous falsifiability; Corollary on simultaneous verifiability and falsifiability; and Theorem on uniqueness of relative negation. The concluding conjecture to map superposition, while tautologous, as (((A∩B)∪(C∩D))|(B∪D))⊃((A|B)∪(C|D)) is replaced with this non-tautologous equation: (((A∩B)∪(C∩D))|(B∩D))⊃((A|B)∩(C|D)).
Category: Set Theory and Logic

[214] viXra:1807.0300 [pdf] submitted on 2018-07-17 15:04:21

Refutation of the Postulate of Contradiction Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The conjecture that a postulate of contradiction for superposition implies contradiction to map quantum logic is refuted.
Category: Set Theory and Logic

[213] viXra:1807.0294 [pdf] submitted on 2018-07-18 00:57:44

Refutation that Classical Logic is a Completion of Quantum Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The distributivity of classical conjunction and disjunction does not fail. The conclusion to translate quantum negation as not necessity is not tautologous. This refutes quantum logic as a fragment of classical logic (or vice versa, as others write).
Category: Set Theory and Logic

[212] viXra:1807.0288 [pdf] submitted on 2018-07-16 21:41:03

Refutation of Quantum Logic as Tautologous Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 5 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

We evaluate quantum logic as described from two sources with different readership levels. We find quantum logic is not tautologous.
Category: Set Theory and Logic

[211] viXra:1807.0279 [pdf] submitted on 2018-07-15 15:01:34

Refutation of Whewell's Axiom of Causality Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

"Everything that becomes or changes must do so owing to some cause; for nothing can come to be without a cause." — Plato in Timaeus William Whewell [1794-1866] refined this as the concept of causality to depend on three axioms: nothing takes place without a cause; the magnitude of an effect is proportional to the magnitude of its cause; and to every action there is an equal and opposed reaction. The second axiom is tautologous, but the others are not, hence refuting the conjecture. From a metaphysical view, the axiom of causality is a bar to miracle because first cause is always assumed. This is overcome with rewriting the conjecture as "The necessity of effect implies the possibility of cause or no cause".
Category: Set Theory and Logic

[210] viXra:1807.0231 [pdf] submitted on 2018-07-12 23:29:05

If at Least One Question Implies Any Answer, Then Any Question Implies at Least One Answer. Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Is it true that any question implies at least one answer? No. Is it true that at least one question implies any answer? No. Is it true that if at least one question implies any answer, then any question implies at least one answer? Yes.
Category: Set Theory and Logic

[209] viXra:1807.0157 [pdf] submitted on 2018-07-09 06:47:49

Shortest Refutation of Gödel's Completeness Theorem Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

"By Gödel's completeness result, the formula (∀x.R(x,x))→(∀x∃y.R(x,y)) holds in all structures, and hence must have a natural deduction proof." We prove the formula is not tautologous, meaning it does not hold in all structures and serves as a contra-example. Hence Gödel's completeness theorem is refuted.
Category: Set Theory and Logic

[208] viXra:1807.0051 [pdf] submitted on 2018-07-02 23:08:38

Definition of Nothing in Mathematical Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Nothing is defined as "not necessarily a thing". This leads to how to collect not everything as nothing in multiple variables into a larger nothing variable, implying a set of nothing as a null set. We write this as nothing in p and nothing in q and nothing in r are all greater than nothing in s: ((~#p&~#q)&~#r)>~#s TTTT TTTT CTTT TTTT. As rendered, this is not tautologous, although nearly so with one deviant C contingency (falsity) value. Hence a collection of nothing does not imply anything outside itself. By extension, the null set is not logically feasible and cannot exist: a collection must contain something even though it is nothing.
Category: Set Theory and Logic

[207] viXra:1806.0450 [pdf] submitted on 2018-06-29 14:56:00

Refutation of the Free Will Hypothesis Based on Its Defective Fin Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Axioms for SPIN and TWIN are tautologous, but the axiom for FIN is not tautologous. Because the assumption of axiom FIN is essential to the authors' proof, the Free Will hypothesis is also not tautologous and refuted by its own derivation. This means the Free Will theorem can not be reasserted by resurrection as such.
Category: Set Theory and Logic

[206] viXra:1806.0449 [pdf] submitted on 2018-06-30 00:31:58

Refutation of the Strong Free Will Hypothesis Based on Its Defective Min' Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

"MIN′: In an A-first frame, B can freely choose any one of the 33 directions w, and a’s prior response is independent of B’s choice. Similarly, in a B-first frame, A can independently freely choose any one of the 40 triples x, y, z, and b’s prior response is independent of A’s choice." The equation as rendered is not tautologous. This means axiom MIN', as replacement for the previous FIN in the Free Will theorem, is not tautologous. Because the assumption of axiom MIN' is essential to the authors' proof, the Strong Free Will theorem is also not tautologous and refuted by its own derivation. This also means the Strong Free Will theorem can not be reasserted by resurrection as such.
Category: Set Theory and Logic

[205] viXra:1806.0338 [pdf] submitted on 2018-06-22 06:37:14

Independency of the Provability of Cliques

Authors: Holger H. Hoo
Comments: 27 Pages.

In this paper, we introduce DLS-MC, a new stochastic local search algorithm for the maximum clique problem. DLS-MC alternates between phases of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, during which vertices of the current clique are swapped with vertices not contained in the current clique. The selection of vertices is solely based on vertex penalties that are dynamically adjusted during the search, and a perturbation mechanism is used to overcome search stagnation. The behaviour of DLS-MC is controlled by a single parameter, penalty delay, which controls the frequency at which vertex penalties are reduced. We show empirically that DLSMC achieves substantial performance improvements over state-of-the-art algorithms for the maximum clique problem over a large range of the commonly used DIMACS benchmark instances.
Category: Set Theory and Logic

[204] viXra:1806.0213 [pdf] submitted on 2018-06-19 14:16:30

Refutation of the Solovay Theorem Copyright © 2008, 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Solovay’s arithmetical completeness theorem for provability logic is refuted by showing the following are not tautologous: Löb's rule as an inference; Gödel's logic system (GL); Gödel's second incompleteness theorem; inconsistency claims of Peano arithmetic (PA); and inability to apply semantical completeness to results which are not contradictory and which are not tautologous.
Category: Set Theory and Logic

[203] viXra:1806.0162 [pdf] submitted on 2018-06-12 23:21:45

The Converse Implication Operator Eqt as a Tense Connective Copyright © 2008, 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 4 Pages. © Copyright 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The converse implication operator named EQT arises to study the inequality in the tense of time for Past > Present > Future. EQT is symmetrically bivalent with the 8-bit pattern {1101 1101} as decimal 187. It is shown that: Past in terms of Present is a falsity; Present in terms of Present is a tautology; and Future in terms of Present is a tautology. Derivations are by Peirce NOR and a 2-tuple truth table. What follows is that tense is not tautologous but non contingent and a truthity. Hence the assumption that time is a theerem is mistaken.
Category: Set Theory and Logic

[202] viXra:1806.0113 [pdf] submitted on 2018-06-09 22:49:48

Refutation of Constructive Brouwer Fixed Point Theorem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-servcies dot com

We construct the Brouwer fixed point theorem (BFPT) as implications of four variables as the antecedent. Because the consequent is composed of disjunctions of ordered pairs, the totality of ordered combinations requires that the argument connective is equivalence. The result is not tautologous and refutes BFPT using a constructive proof. (If the consequent is taken as a multiplicity of ordered combinations, the equivalence connective and the implication connective share the same table result which deviates further from tautology.) We conclude that BFPT is mislabeled as a theorem, as non constructively based on set theory, and correctly named as the Brouwer fixed point conjecture (BFPC).
Category: Set Theory and Logic

[201] viXra:1806.0109 [pdf] submitted on 2018-06-08 12:57:27

Evaluating f(x) = C for Infinite Set Domains

Authors: Ron Ragusa
Comments: 5 Pages. email: ron.ragusa@gmail.com

In a previous paper, The Function f(x) = C and the Continuum Hypothesis, posted on viXra.org (viXra:1806.0030), I demonstrated that the set of natural numbers can be put into a one to one correspondence with the set of real numbers, f : N → R. In that paper I used the function f(x) = C to create an indexed array of the function’s real number domain d, the constant range, C, and the index value of each iteration of the function’s evaluation, i, for each member of the domain di. The purpose of the exercise was to provide constructive proof of Cantor’s continuum hypothesis which has been shown to be independent of the ZFC axioms of set theory. Because the domain of f(x) = C contains all real numbers, evaluating and indexing the function over the entire domain leads naturally to the bijective function f : N → R. In this paper I’ll demonstrate how the set of natural numbers N can be put into a one to one correspondence the power set of natural numbers, P(N). From this I will derive the bijective function f : N → P(N). Lastly, I’ll propose a conjecture asserting that f(x) = C can be employed to construct a one to one correspondence between the natural numbers and any infinite set that can be cast as the domain of the function.
Category: Set Theory and Logic

[200] viXra:1806.0030 [pdf] submitted on 2018-06-03 07:41:10

The Function f(x) = C and the Continuum Hypothesis

Authors: Ron Ragusa
Comments: 5 Pages.

This paper examines whether or not an analysis of the behavior of the continuous function f(x) = C, where C is any constant, on the interval (a, b) where a and b are real numbers and a < b, will provide a method of proving the truth or falsity of the CH. The argument will be presented in three theorems and one corollary. The first theorem proves, by construction, the countability of the domain d of f(x) = C on the interval (a, b) where a and b are real numbers. The second theorem proves, by substitution, that the set of natural numbers N has the same cardinality as the subset of real numbers S on the given interval. The corollary extends the proof of theorem 2 to show that N and R are of the same cardinality. The third theorem proves, by logical inference, that the CH is true.
Category: Set Theory and Logic

[199] viXra:1805.0543 [pdf] submitted on 2018-05-31 13:58:46

Refutation of the Löwenheim–Skolem Theorem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services.com

Refuted in each segment and in toto.
Category: Set Theory and Logic

[198] viXra:1805.0332 [pdf] submitted on 2018-05-19 00:21:07

Set Theory

Authors: Salvatore Gerard Micheal
Comments: 4 Pages.

Objective: to create a precise axiomatic system of exact connected meaningful statements which we can use to build set theory from. The purpose is to make explicit our notion of elements before that notion is used in set equivalence. It is required in a dependency framework. We will find that this system is most economical in terms of repetition and symbol usage. We will state and prove a theorem regarding this.
Category: Set Theory and Logic

[197] viXra:1805.0327 [pdf] submitted on 2018-05-17 13:04:37

On The Mistake That Is Riemann's Prime Zeta Function

Authors: Max Davidović
Comments: 1 Page.

I have succesfully disproved Riemann's hypothesis. It has taken me years of dedicated reseach to realise that such and endeavor is frankly impossible without changing the axioms upon which math is built. The file that I have attached is a shitpost born from my frustration and anger.
Category: Set Theory and Logic

[196] viXra:1805.0302 [pdf] submitted on 2018-05-15 22:37:07

Refutation of Gettier Problem of Justified True/false Belief © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

We find justified true belief is not a theorem. We find justified false belief is also not a theorem. This means the Gettier problem as the superset of the justified belief arguments is refuted as a problem and resolved as a non-problem.
Category: Set Theory and Logic

[195] viXra:1805.0266 [pdf] submitted on 2018-05-13 22:32:32

Many Questions and Many Answers © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services.com

If for some questions there are many answers, then for many questions there are some answers, this is a theorem. If for many questions there are some answers, then for some questions there are many answers, this is not a theorem but a truthity.
Category: Set Theory and Logic

[194] viXra:1805.0249 [pdf] submitted on 2018-05-12 13:16:02

Refutation of the Continuum Hypothesis © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

"The continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers." That equation is not tautologous, and the briefest known refutation of the continuum hypothesis.
Category: Set Theory and Logic

[193] viXra:1805.0223 [pdf] submitted on 2018-05-12 04:55:58

Refutation of the Molyneux Problem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The equation as rendered is not tautologous. This means the Molyneux problem is resolved as two unrelated states, and hence not a problem.
Category: Set Theory and Logic

[192] viXra:1805.0215 [pdf] submitted on 2018-05-10 23:50:55

Refutation of the Hilbert Grand Hotel Paradox © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The equation as rendered is not contradictory but rather falsity. Hence this refutes the Hilbert Grand Hotel paradox. Remark: We could not reduce this paradox to one variable because rooms and guests are distinctly counted.
Category: Set Theory and Logic

[191] viXra:1805.0189 [pdf] submitted on 2018-05-09 12:10:44

Method of Reducing Paradox to not Contradictory and in One Variable © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

This demonstration serves as two examples to confirm the method that a paradox refuted as not contradictory is also reducible to one variable. The examples are the paradox of Zhuangzi known as the butterfly dream and the paradox of Maimonides known as the argument from freewill.
Category: Set Theory and Logic

[190] viXra:1805.0183 [pdf] submitted on 2018-05-09 20:19:28

Refutation of Fredkin Paradox in One Variable © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

"The more equally attractive two alternatives seem, the harder it can be to choose between them —- no matter that, to the same degree, the choice can only matter less." This is not contradictory, and not tautologous, thereby refuting the Fredkin paradox, and in one variable p as chosen state and alternative state.
Category: Set Theory and Logic

[189] viXra:1805.0158 [pdf] submitted on 2018-05-06 21:33:10

Refutation of Fitch's Paradox of Knowability © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

We find the Fitch paradox to be tautologous, that is, not contradictory, and hence a theorem. We state the theorem as "every truth is discernible", and we add, "by the instant modal logic model checker". A subsequent invocation of Gödel incompleteness and with jettison of the knowability rule to generalize and solve the paradox is contradictory, proving nothing.
Category: Set Theory and Logic

[188] viXra:1805.0155 [pdf] submitted on 2018-05-06 23:17:06

Refutation of Drinker's Paradox © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The drinker's paradox is not tautologous and also not contradictory. Therefore, the drinker's paradox is refuted as a paradox.
Category: Set Theory and Logic

[187] viXra:1805.0150 [pdf] submitted on 2018-05-07 07:40:33

Refutation of the New Riddle of Induction © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Both states of the argument are tautologous, not contradictory, and hence theorems. Therefore the new riddle of induction is refuted as a riddle or paradox.
Category: Set Theory and Logic

[186] viXra:1805.0146 [pdf] submitted on 2018-05-07 10:09:55

Refutation of the Paradox of Epimenides the Cretan © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The argument is: "Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie?" It is not tautologous and not contradictory. Therefore the paradox of Epimenides is refuted as a paradox. The answer to the question "Was this a lie" is neither contradiction nor proof.
Category: Set Theory and Logic

[185] viXra:1805.0138 [pdf] submitted on 2018-05-07 23:05:53

Refutation of the Paradox of Moses Maimonides © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Does God know or does He not know that a certain individual will be good or bad? If thou sayest 'He knows', then it necessarily follows that the man is compelled to act as God knew beforehand he would to act, otherwise God's knowledge would be imperfect. The question is tautologous. The additional sentence with the first is not tautologous, not contradictory. Therefore the paradox of Maimonides is refuted as a paradox.
Category: Set Theory and Logic

[184] viXra:1805.0124 [pdf] submitted on 2018-05-05 23:01:52

Refutation of the Prisoner Paradox© Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The three states of the prisoner paradox are not contradictory. This means the prisoner paradox is not a paradox.
Category: Set Theory and Logic

[183] viXra:1805.0110 [pdf] submitted on 2018-05-05 08:48:59

Refutation of Carroll's Tortoise and Achilles as a Paradox © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

A: "Things that are equal to the same are equal to each other" B: "The two sides of this triangle are things that are equal to the same" Therefore Z: "The two sides of this triangle are equal to each other" The argument as rendered is tautologous and hence a theorem. Because it is not contradictory: this refutes Carroll's tortoise and Achilles as a paradox.
Category: Set Theory and Logic

[182] viXra:1805.0106 [pdf] submitted on 2018-05-05 11:23:45

Refutation of the Paradox of the Concept of an Analysis as Both Correct and Informative © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The concept of an analysis as both correct and informative is not a paradox.
Category: Set Theory and Logic

[181] viXra:1805.0105 [pdf] submitted on 2018-05-05 14:58:45

Refutation of the Newcomb Paradox© Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The two separate states of the Newcomb paradox are not contradictory (and not tautologous), so the paradox is refuted. However interestingly if the states are tested as one system (either or), it is tautologous.
Category: Set Theory and Logic

[180] viXra:1804.0351 [pdf] submitted on 2018-04-25 09:48:21

Refutation of Additive Arithmetic Operations in the Riemann Sphere © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The definition of multiplication for extended complex numbers and undefined values of ∞–∞ and 0×∞ are theorems. The definition of addition for extended complex numbers is not a theorem. The custom of forcing a field definition for extended complex numbers is mistaken as are the undefined values of the quotients 0/0 and ∞/∞.
Category: Set Theory and Logic

[179] viXra:1804.0174 [pdf] submitted on 2018-04-12 13:53:22

The Universe Equation

Authors: Seamus McCelt
Comments: 2 Pages.

If you claim there are particles: there would actually have to be particles.
And that would mean there are about 18 different microscopic things that work flawlessly together -- just like clockwork to make even just one basic atom "gear" set.
If you have larger sized atoms: it would be like throwing more and more gear sets into the clockwork -- but that is ok because no matter what you throw in -- it will still work just fine.
How can an infinity of 18 different things (infinity times 18 different things) just happen to be here, know how work together as a group and also successfully work together as a group(s)?
How is that possible? It isn't...

Stuff cannot be made from what they call "particles."

If there are particles; this is equation of the universe:
Universe = Infinity × {a,b,c,d,f,g,h,j,k,l,m,o,p,q,t,w,x,y,z}
Category: Set Theory and Logic

[178] viXra:1803.0318 [pdf] submitted on 2018-03-19 18:43:19

Refutation of Abductive Reasoning © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Abductive logic of C.S. Peirce is refuted as not tautologous.
Category: Set Theory and Logic

[177] viXra:1803.0180 [pdf] submitted on 2018-03-12 15:52:51

Refutation of the Euathlus Paradox: Neither Pay© Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved.

Regardless of who wins the lawsuit of Portagoras, Euathlus does not pay. Hence the Euathlus paradox is refuted and resolved by default in favor of Euathlus.
Category: Set Theory and Logic

[176] viXra:1803.0094 [pdf] submitted on 2018-03-07 11:14:13

Refutation of Cantor's Original Continuum Hypothesis Via Injection and Binary Trees © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

This is the briefest known such refutation of Cantor's continuum conjecture.
Category: Set Theory and Logic

[175] viXra:1803.0088 [pdf] submitted on 2018-03-07 03:33:06

The Continuum Hypothesis

Authors: Chris Pindsle
Comments: 12 Pages.

A proof of the Continuum Hypothesis as originally posed by Georg Cantor in 1878; that an uncountable set of real numbers has the same cardinality as the set of all real numbers. Any set of real numbers can be encoded by the infinite paths of a binary tree. If the binary tree has an uncountable node it must have a descendant with 2 uncountable successors. Each of those will have descendants with 2 uncountable successors, recursively. As a result the infinite paths of an uncountable binary tree will have the same cardinality as the set of all real numbers, as will the uncountable set of real numbers encoded by the tree.
Category: Set Theory and Logic

[174] viXra:1803.0034 [pdf] submitted on 2018-03-02 17:28:55

Meth8/VŁ4 on Complex Numbers (ℂ) © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

Meth8/VŁ4 maps complex numbers (ℂ) using implication and not equivalence which serves to reason since complex numbers are imaginary and not real.
Category: Set Theory and Logic

[173] viXra:1802.0357 [pdf] submitted on 2018-02-25 07:56:34

P=NP Resolution, with 3-Sat not Tautologous © Copyright 2017-2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. © Copyright 2017-2018 by Colin James III All rights reserved.

We use the definition of NP as “nondeterministic polynomial time” from Stephen Cook at claymath.org/sites/default/files/pvsnp.pdf. It is not tautologous and is presented in a truth table on 1024-values. An example from the same source for the 3-SAT test as NP-complete for the expression (p∨q∨r)∧(~p∨q∨~r)∧(p∨~q∨s)∧(~p∨~r∨~s), with τ(P)=τ(Q)=Tautologous and τ(R)=τ(S)=contradictory, is not tautologous.
Category: Set Theory and Logic

[172] viXra:1802.0329 [pdf] submitted on 2018-02-22 14:00:52

Refutation of the ef-Axiom © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

The EF-axiom describes the Efremovifč proximity δ by V.A. Efremovič from 1934 and published in Russian in 1951. We interpret the operator δ to mean "nearby" or "in proximity". The size of an antecedent or consequent is not stated for the operator, so we determine that the operator applies to unrelated literals as as ((A∈ B) Nor (B∈ A)). The proof result is for 256-values because four theorems are evaluated as variables. We find the EF-axiom is not tautologous. The implications to topology are legion.
Category: Set Theory and Logic

[171] viXra:1802.0172 [pdf] submitted on 2018-02-14 06:16:03

Refutation of Tarski's Undefinability of Truth Theorem © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

This refutes Tarski's theorem for the undefinability of truth as: "no definable and sound extension of Peano Arithmetic can be complete"; or in abstract terms, "the proof of a system cannot be demonstrated by itself". Tarski's theorem is an arguable equivalent to Godel's incompleteness theorem, as based on the liar's paradox. [Remark added later: Tarski's theorem as used since about 1936 is an underpinning of quantum theory and a universal justification for atheism.]
Category: Set Theory and Logic

[170] viXra:1801.0380 [pdf] submitted on 2018-01-27 22:55:45

Refutation of the Direct Correspondence of Quantum Gates to Reversible Classical Gates © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

These quantum gates do not directly correspond to reversible classical gates: CNOT (XOR, Feynman); Toffoli (AND); X (NOT); and n-qubit Toffoli (AND). Hence quantum gates cannot map to bivalent logic. This paper demonstrates the shortest refutation.
Category: Set Theory and Logic

[169] viXra:1801.0253 [pdf] submitted on 2018-01-19 16:14:43

Refutation of Neutrosophic Soft Lattice Theory © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. Papers by and relating to Florentin Smaradnache span many areas at viXra, but the appropriate field is Set Theory and Logic. Many of them are translated as crude ruse to garner publicity. The study neutrosohic logic is mostly specious.

We evaluate the neutrosophic logic based on its most atomic level of soft latices, as published by Springer-Verlag in 2016. We refute the theorem "Every neutrosophic soft lattice is a one-sided distributive neutrosophic soft lattice." This brief evaluation implies that the field of soft set theory as originally introduced by D. Molodtsov is suspicious and specifically that the field of neutrosophic logic, as evidenced in its basis of soft set theory, is unworkable. This conclusion is multitudinal because of the plethora of duplicated papers as translations in multiple fields at vixra.org regarding the neutrosophic logic system of Florentin Smarandache.
Category: Set Theory and Logic

[168] viXra:1801.0122 [pdf] submitted on 2018-01-11 01:52:00

The Incalculable Continuum

Authors: Miguel A. Sanchez-Rey
Comments: 1 Page.

A continuum in sight.
Category: Set Theory and Logic

[167] viXra:1712.0600 [pdf] submitted on 2017-12-26 09:29:23

ZF Law of Excluded Middle on Infinite Sets (Lemi) © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved

Eqs. 1.2 and 1.4 as rendered are not tautologous. Hence Meth8/VL4 finds LEMI suspicious.
Category: Set Theory and Logic

[166] viXra:1712.0595 [pdf] submitted on 2017-12-26 00:23:47

The Formal-Logical Analysis of the Foundation of Set Theory

Authors: Temur Z. Kalanov
Comments: 17 Pages.

The critical analysis of the foundation of set theory is proposed. The unity of formal logic and rational dialectics is the correct methodological basis of the analysis. The analysis leads to the following results: (1) the mathematical concept of set should be analyzed on the basis of the formal-logical clauses “Definition of concept”, “Logical class”, “Division of concept”, “Basis of division”, “Rules of division”; (2) the standard mathematical theory of sets is an erroneous theory because it does not contain definition of the concept “element (object) of set”; (3) the concept of empty set (class) is a meaningless, erroneous, and inadmissible one because the definition of the concept “empty set (class)” contradicts to the definition of the logical class. (If the set (class) does not contain a single element (object), then there is no feature (sign) of the element (object). This implies that the concept of empty set (class) has no content and volume (scope). Therefore, this concept is inadmissible one); (4) the standard mathematical operations of union, intersection and difference of sets (classes) are meaningless, erroneous and inadmissible operations because they do not satisfy the following formal-logical condition: every separate element (object) of the set (class) must be in only one some set (class) and cannot be in two sets (classes). Thus, the results of formal-logical analysis prove that the standard mathematical theory of sets is an erroneous theory because it does not satisfy the criterion of truth.
Category: Set Theory and Logic

[165] viXra:1712.0403 [pdf] submitted on 2017-12-13 04:09:51

There is no Standard Model of ZFC

Authors: Jaykov Foukzon
Comments: 15 Pages.

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st), (ii) let k be an inaccessible cardinal then  ~Con(ZFC+∃k),[10]-[11].
Category: Set Theory and Logic

[164] viXra:1712.0386 [pdf] submitted on 2017-12-12 01:26:41

On Multi-Criteria Pythagorean Fuzzy Decision-Making

Authors: Liguo Fei, Yong Deng
Comments: 21 Pages.

Pythagorean fuzzy set (PFS) initially extended by Yager from intuitionistic fuzzy set (IFS), which can model uncertain information with more general conditions in the process of multi-criteria decision making (MCDM). The fuzzy decision analysis of this paper is mainly based on two expressions in Pythagorean fuzzy environment, namely, Pythagorean fuzzy number (PFN) and interval-valued Pythagorean fuzzy number (IVPFN). We initiate a novel axiomatic definition of Pythagorean fuzzy distance measure, including PFNs and IVPFNs, and put forward the corresponding theorems and prove them. Based on the defined distance measures, the closeness indexes are developed for PFNs and IVPFNs inspired by the idea of technique for order preference by similarity to ideal solution (TOPSIS) approach. After these basic definitions have been established, the hierarchical decision approach is presented to handle MCDM problems under Pythagorean fuzzy environment. To address hierarchical decision issues, the closeness index-based score function is defined to calculate the score of each permutation for the optimal alternative. To determine criterion weights, a new method based on the proposed similarity measure and aggregation operator of PFNs and IVPFNs is presented according to Pythagorean fuzzy information from decision matrix, rather than being provided in advance by decision makers, which can effectively reduce human subjectivity. An experimental case is conducted to demonstrate the applicability and flexibility of the proposed decision approach. Finally, the extension forms of Pythagorean fuzzy decision approach for heterogeneous information are briefly introduced as the further application in other uncertain information processing fields.
Category: Set Theory and Logic

[163] viXra:1712.0368 [pdf] submitted on 2017-12-09 22:30:01

The Brain Simulator Reply (BSR) of the Chinese Room Argument (Cra) is Confirmed. © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

"[O]one cannot infer from X simulates Y, and Y has property P, to the conclusion that therefore X has Y's property P for arbitrary P." is tautologous. "The contrapositive of the inference is logically equivalent—X simulates Y, X does not have P therefore Y does not [have P]" is not tautologous. The two conjectuses are not logically equivalent. Hence the brain Simulator reply of the Chinese room argument is confirmed and validated.
Category: Set Theory and Logic

[162] viXra:1712.0205 [pdf] submitted on 2017-12-06 16:42:00

Refutation of Cantor's Diagonal Argument © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We map the argument for Cantor's diagonal argument into the Meth8 modal logic model checker. The two main equations as rendered are not tautologous. Hence Cantor's diagonal argument is not supported.
Category: Set Theory and Logic

[161] viXra:1712.0204 [pdf] submitted on 2017-12-06 17:11:12

Refutation of Axiom of Choice Via Refutation of the Gödel-Löb Axiom © Copyright 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2016 by Colin James III All rights reserved.

Refutation of the Gödel-Löb Axiom implies the refutation of the Axiom of Choice. Both axioms as separately rendered are also not tautologous.
Category: Set Theory and Logic

[160] viXra:1712.0139 [pdf] submitted on 2017-12-06 12:44:00

A Proof of the Falsity of the Axiom of Choice.

Authors: Johan Noldus
Comments: 1 Page.

We show that the axiom of choice is false.
Category: Set Theory and Logic

[159] viXra:1711.0473 [pdf] submitted on 2017-11-29 13:25:59

Unification by Neutrosophic Logic not Tautologous © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2017 by Colin James III All rights reserved.

We evaluate the two axioms with attendant rules which define neutrosophic logic. Both produce the same proof table which is not tautologous. What follows is that neutrosophic logic is not bivalent, but a vector space, and cannot unify other logics in a tautology.
Category: Set Theory and Logic

[158] viXra:1711.0462 [pdf] submitted on 2017-11-28 22:51:13

Definition of the Zero Knowledge Proof Refuted © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

The formal definition of the zero-knowledge proof as rendered is not tautologous. What follows is the assumption that in NP all problems and all languages have zero-knowledge proofs is mistaken. What also follows is that one-way functions do not exist.
Category: Set Theory and Logic

[157] viXra:1711.0425 [pdf] submitted on 2017-11-25 13:35:07

Refutation of Realizability Semantics for QML © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages.

We found that the following are not tautologous: Epistemic Church's Thesis; EZF induction schema; and Scedrov's modal foundation. We did not test subsequent axioms. What follows is that Flagg's construction, Goodman's intensional set theory, and epistemic logic are suspicious.
Category: Set Theory and Logic

[156] viXra:1711.0416 [pdf] submitted on 2017-11-25 11:03:31

Stone Space Type Lattice Logic Model © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

Injection of a binary relation does not support it as symmetric and irreflexive.
Category: Set Theory and Logic

[155] viXra:1711.0412 [pdf] submitted on 2017-11-24 21:11:36

Refutation of the Universal Finite Set © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We evaluated two parts of the proof of Lemma 2 (Folklore). The equations as rendered are not tautologous.
Category: Set Theory and Logic

[154] viXra:1711.0378 [pdf] submitted on 2017-11-20 08:04:42

Refutation of Chaitin's Theorem of Incompleteness © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages.

We evaluate Chaitin's incompleteness theorem of 1974. Martin Davis described it as “a dramatic extension of Gödel’s incompleteness theorem” in 1978. We find the approach of the conjecture is moot, refute Chaitin's theorem of incompleteness, and remark that Chaitin's constant is suspicious.
Category: Set Theory and Logic

[153] viXra:1711.0364 [pdf] submitted on 2017-11-19 10:38:24

The Shortest Refutation of Gödel's Theorem of Incompleteness © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2017 by Colin James III All rights reserved.

Our example in the positive and contra example show the shortest known definitive refutation for Gödel's incompleteness theorem.
Category: Set Theory and Logic

[152] viXra:1711.0357 [pdf] submitted on 2017-11-18 08:01:42

Note on Grilliot's Trick and Effective Implication © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We evaluate two of the simpler equations for tautology which relate to Grilliot's trick and the standard extensionality trick via effective implication. Eqs., as rendered, are not tautologous. According to variant sysem VŁ4, this means Grilliot's trick, effective implication, and the subsequent non standard extensionality trick are not bivalent, but rather are a vector space.
Category: Set Theory and Logic

[151] viXra:1711.0323 [pdf] submitted on 2017-11-15 14:07:02

Logic not Tautologous in Neutrosophic Sets © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We test a theorem and two properties from above. Eqs. 1.1, 5.2, and 6.1 should be tautologous, but are not.
Category: Set Theory and Logic

[150] viXra:1711.0320 [pdf] submitted on 2017-11-15 18:23:23

Law of Self-Equilibrium: not Law; not Paradox © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

The law of self-equilibrium sometimes uses this example: Too much work produces sickness; sickness produces less work; therefore, too much work implies less work. We rewrite the sentence to replace the connective verb with "causes" for better meaning and also include a modal operator for clarity: Too much work causes possible sickness; sickness causes less work; therefore, too much work causes less work. ... the law of self-equilibrium is not tautologous, and hence not a theorem and not a paradox.
Category: Set Theory and Logic

[149] viXra:1711.0312 [pdf] submitted on 2017-11-15 06:04:19

Re: Deducibility Theorems in Boolean Logic Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, Usa e-Mail: Smarand@unm.edu Http://vixra.org/abs/1003.0171

Authors: Colin James III
Comments: 1 Page.

As presumably a basis for neutrosophic logic these mistakes were found: Theorems 1 and 2 are not tautologous.
Category: Set Theory and Logic

[148] viXra:1711.0290 [pdf] submitted on 2017-11-12 12:05:19

Rationale of Rendering Quantifiers as Modal Operators © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 14 Pages. © 2016 by Colin James III All rights reserved.

The rationale for rendering quantifiers as modal operators in Meth8 has arguments from satisfiability and reproducability of invalidating and validating syllogisms. The Square of Opposition (original) produced four combinations for each corner A, I, E, O for 4 ^ 4 = 256 syllogisms. Medieval scholars determined 24 of the 256 syllogisms were valid deductions. Of those, 9 were made valid but only after additional known assumptions were applied as fix ups. Meth8 Tautologous none of the 24 syllogisms before fix ups. Meth8 also discovered correct additional assumptions to render the other 15 syllogisms Tautologous. We use Meth8 to replicate the 24 valid syllogisms derived from the original Square of Opposition. In the process we make three recent advances. 1. A third assumption is needed to fix up Modus Cesare EAE-2 2. The third assumption cannot be removed from Modus Camestros AEO-2 (as in other syllogisms with known third assumptions); and 3. No third assumptions are required for the other 22 syllogisms.
Category: Set Theory and Logic

[147] viXra:1711.0288 [pdf] submitted on 2017-11-12 12:11:20

Square of Opposition Modern Revised: not Validated as Tautologous © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © 2016 by Colin James III All rights reserved.

The modern, revised square of opposition is not validated as tautologous by the Meth8 logic checker in five models for all expressions. This leads us to consider that any logic system based on the square of opposition is spurious. What follows then is that a first order predicate logic based on the square of opposition is now suspicious.
Category: Set Theory and Logic

[146] viXra:1711.0113 [pdf] submitted on 2017-11-03 07:20:31

Tautology Problem and Two Dimensional Formulas

Authors: Deniz Uyar
Comments: 38 Pages.

Finding whether a boolean formula is a tautology or not in a feasible time is an important problem of computer science. Many algorithms have been developed to solve this problem but none of them is a polynomial time algorithm. Our aim is to develop an algorithm that achieve this in polynomial time. In this article, we convert boolean functions to some graph forms in polynomial time. They are called two dimensional formulas and similar to AND-OR graphs except arches on them are bidirectional. Then these graphs are investigated to find properties which can be used to differentiate tautological formulas from non tautological ones.
Category: Set Theory and Logic

[145] viXra:1710.0309 [pdf] submitted on 2017-10-28 12:24:53

Alpha-Conversion for Lambda Terms with Explicit Weakenings

Authors: George Cherevichenko
Comments: 15 Pages.

Using explicit weakenings, we can define alpha-conversion by simple equations without any mention of free variables.
Category: Set Theory and Logic

[144] viXra:1710.0237 [pdf] submitted on 2017-10-21 10:32:06

Fundamental Set Theory

Authors: Anders Lindman
Comments: 3 Pages.

The fundamental set theory (FST) is defined as an axiomatic set theory using nonclassical three-valued logic in the foundation and classical two-valued logic in its applications. In this way the nonclassical logic becomes encapsulated and is only used for resolving inconsistencies such as Russell's paradox.
Category: Set Theory and Logic

[143] viXra:1710.0226 [pdf] submitted on 2017-10-20 08:20:23

Do The Real Numbers form a Set?

Authors: Divyendu Priyadarshi
Comments: 2 Pages.

I have argued that the Real Numbers do not form a set in the sense that they lack any specific character to define them. I am not a professional mathematician but a Physics teacher. So my arguments may lack mathematical precision. All suggestions and criticisms are heartily welcome.
Category: Set Theory and Logic

[142] viXra:1710.0223 [pdf] submitted on 2017-10-19 23:09:13

The Set of Standard Numbers

Authors: Anders Lindman
Comments: 3 Pages.

The set of standard numbers D is constructed from an axiom of infinite fraction sum together with the power set of the set of all rational numbers in the form 2^-n. The power set contains all possible infinite binary sequences who represent the fraction part of the standard numbers together with the integers for the whole number part. The set of standard numbers includes the rational numbers and forms a field (D, +, *) similar to, yet distinct from the set of real numbers R.
Category: Set Theory and Logic

[141] viXra:1710.0035 [pdf] submitted on 2017-10-04 01:50:21

The Statistical Proof of the Riemann Hypothesis

Authors: Dmitri Martila
Comments: 4 Pages.

Derived the Statistics of the un-solved problems (conjectures). The probability, what a conjecture will be solved is 50 %. The probability, that a conjecture is true is p=37 %. The probability, what we get to know the latter is psi=29 %....
Category: Set Theory and Logic

[140] viXra:1709.0391 [pdf] submitted on 2017-09-26 10:31:21

A New Axiom for ZFC Set Theory that Results in a Problem

Authors: Andew Banks
Comments: 5 Pages.

This article adds a new axiom to ZFC that assumes there is a set x which is initially the empty set and thereafter the successor function (S) is instantly applied once in-place to x at each time interval (½ⁿ n>0) in seconds. Next, a very simple question is proposed to ZFC. What is x after one second elapses? By definition, each time S is applied in-place to x, a new element is inserted into x. So, given that S is applied at each time interval (½ⁿ n>0) then an infinite collection of elements is added to x so, x is countable infinite. On the other hand, since x begins as the empty set and only S is applied to x then x cannot be anything other than a finite natural number. Hence, x is finite. Clearly, in-place counting according to the interval timings (½ⁿ n>0) demonstrates a problem in ZFC
Category: Set Theory and Logic

[139] viXra:1709.0076 [pdf] submitted on 2017-09-07 11:08:35

P Versus Np's Answer.

Authors: Bhasanpal S.
Comments: Pages.

Always think simple to answer a question which is being seem tough.
Category: Set Theory and Logic

[138] viXra:1708.0156 [pdf] submitted on 2017-08-14 10:23:58

Some Critical Notes on the Cantor Diagonal Argument

Authors: Philip Molyneux
Comments: 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in set theory to draw a distinction between the cardinality of the natural numbers and that of the real numbers. In the absence of a verified English translation of the original 1891 Cantor paper from which it is said to be derived, the CDA is discussed here using a consensus from the forms found in a range of published sources (from "popular" to "professional"). Some general comments are made on these sources. The discussion then focusses on the CDA as applied to the correspondence between the set of the natural numbers, and the set of real numbers in the open range (0,1) in their expansion from decimal digits (0.123… etc.). Four points critical of the CDA are raised: (1) The conventional presentation of the CDA forms a putative new real number (X) from the "diagonal" of the chosen list of real numbers and which is therefore not on this initial list; however, it omits to consider that it may indeed be on the later part of the list, which is never exhausted however far the "diagonal" list is extended. (2) This aspect, combined with the fact that X is still composed of decimal digits, that is, it is a real number as defined, indicates that it must be on the later part of the list, that is, it is not a "new" number at all. (3) The conventional application of the CDA leads to one putative "new" real number (X); however, the logical extension of this in its "exhaustive" application, that is, by using all possible different methods of alteration of the decimal digits on the "diagonal", and by reordering the list in all possible ways, leads to a list of putative "new" real numbers that become orders of magnitude longer than the chosen "diagonal" list. (4) The CDA is apparently considered to be a method that is applicable generally; however, testing this applicability with the natural numbers themselves leads to a contradiction. Following on from this, it is found that it indeed is possible to set up a one-to-one correspondence between the natural numbers and the real numbers in (0,1), that is, ! ⇔ "; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension of the natural number is intended to be a mirror image of the left hand extension of the real number. It is also shown how this may be extended to real numbers outside the range (0,1). Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998 critical review of "hopeless papers" dealing with the CDA; this form is also examined from the same viewpoints, and to the same conclusions. Finally, some comments are made on the concept of "infinity", pointing out that to consider this as an entity is a category error, since it simply represents an absence, that is, the absence of a termination to a process.
Category: Set Theory and Logic

[137] viXra:1707.0220 [pdf] submitted on 2017-07-16 10:14:11

Proof of ZFC Axioms as Normal Statements No. 2.2

Authors: Thomas Limberg
Comments: 6 Pages. Language: German

We interpret 5 of the 10 axioms of ZFC (Zermelo-Fraenkel set theory with axiom of choice) as normal statements and proof them. So these 5 sentences don't need to be introduced as axioms, but can be used as proven statements.
Category: Set Theory and Logic

[136] viXra:1706.0321 [pdf] submitted on 2017-06-12 04:00:23

Aristotle’s Answer on Russell’s Paradox

Authors: Nikolay Dementev
Comments: 9 Pages.

An attempt of resolving Russell’s paradox with the help of Aristotle’s ideas is presented.
Category: Set Theory and Logic

[135] viXra:1705.0226 [pdf] submitted on 2017-05-14 17:15:50

Is Classical Mathematics Appropriate for Theory of Computation?

Authors: Farzad Didehvar
Comments: 9 Pages.

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”, first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved. .
Category: Set Theory and Logic

[134] viXra:1705.0173 [pdf] submitted on 2017-05-10 12:07:14

Dialectical Logic – Negation Of Classical Logic

Authors: Ilija Barukčić
Comments: 23 Pages. Copyright © 2017 by Ilija Barukčić, Jever, Germany. Published by:

The division of zero by zero turns out to be a long lasting and not ending puzzle in mathematics and physics. An end of this long discussion is not in sight. In particular zero divided by zero is treated as indeterminate thus that a result cannot be found out. It is the purpose of this publication to solve the problem of the division of zero by zero while relying on the general validity of classical logic. According to classical logic, zero divided by zero is one.
Category: Set Theory and Logic

[133] viXra:1704.0161 [pdf] submitted on 2017-04-13 03:51:02

The Number Density Paradox

Authors: Ken Seton
Comments: 6 Pages. Enjoy

In the world of transfinite cardinality, any talk of number density, dart-board hits, proportions or probability is just a pouring from the empty into the void. Transfinite cardinality is unrelated to any normal concept of proportion or density over an interval and no understanding of it can obtained from probabilistic analogies. This is demonstrated by comparing extremely sparse and extremely dense collections of reals which are normally understood as uncountable and countable respectively. Should this incline one to think that Cantor’s equivalence definition is inappropriate for identifying the “size” of an infinite collection ?
Category: Set Theory and Logic

[132] viXra:1704.0115 [pdf] submitted on 2017-04-10 01:13:52

The Simple Infinite Set

Authors: Ken Seton
Comments: 8 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (to critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we observe the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite . Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be. Set theory found itself into some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.
Category: Set Theory and Logic

[131] viXra:1703.0113 [pdf] submitted on 2017-03-13 04:06:52

Failure of the Diagonal Argument

Authors: Wolfgang Mückenheim
Comments: 3 Pages.

It is shown that Cantor's diagonal argument fails because either there is no actual infinity and hence no defined diagonal number or there is actual infinity but the diagonal number cannot be distinguished from all real numbers of the Cantor list. Further it is shown by another argument that there are not uncountably many paths in the complete infinite Binary Tree.
Category: Set Theory and Logic

[130] viXra:1703.0112 [pdf] submitted on 2017-03-13 04:15:07

Set Theory or Slipper Animalcule: Who Wins?

Authors: Wolfgang Mückenheim
Comments: 3 Pages.

Limits of sequences of sets required to define infinite bijections do not only raise paradoxes but cause self-contradictory results.
Category: Set Theory and Logic

[129] viXra:1703.0032 [pdf] submitted on 2017-03-03 16:07:51

The Union is not the Limit.

Authors: W. Mückenheim
Comments: 5 Pages.

Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of sets differ. We will show this for the set of natural numbers by the newly devised powerful tool of arithmogeometry.
Category: Set Theory and Logic

[128] viXra:1702.0293 [pdf] submitted on 2017-02-24 03:55:26

Not Enumerating All Positive Rational Numbers

Authors: W. Mückenheim
Comments: 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.
Category: Set Theory and Logic

[127] viXra:1701.0564 [pdf] submitted on 2017-01-22 04:25:16

The Declaration of Unity and Union

Authors: Amir Deljoo
Comments: 11 Pages.

This is a declaration. The identity of mathematics and number theory or arithmetics. I have defined a pattern here that shows consciousness is a pure unique entity that is present everywhere and whole the existence is a graphical manifestation that has been phenomenoned over to enclose it and I hermetically simplify my intuition to transfer it to curious ones. Since explaining the methodology requires in thousands of pages, the final concluded statements and equations are only declared here.
Category: Set Theory and Logic

[126] viXra:1701.0563 [pdf] submitted on 2017-01-22 04:47:06

بیانیه ی یگانه و انجمن

Authors: امیر دلجو
Comments: 12 Pages.

این سند یک منشور است برای بیان ماهیت ریاضیات و نظریه ی اعداد یا حساب. در اینجا من الگویی را تعریف کرده ام که نشان می دهد، خودآگاهی یک وجود واحده بسیط و همه جاحاضر بوده و هستی به مثابه یک کلیّت، یک تجلّی گرافیکی ست که در جهت افشای این خودآگاهی عارض شده و من هرمس وار، شهود خود بر یگانگی و آفرینش را برای انتقال به انسان های کنجکاو ساده سازی و تحریر کرده ام. از آنجا که تبیین روش شناختی این منشور مستلزم هزاران صفحه است، در اینجا تنها عبارات و معادلات منتج شده ی نهایی اعلان می شود.
Category: Set Theory and Logic

Replacements of recent Submissions

[203] viXra:1911.0412 [pdf] replaced on 2019-12-07 07:06:36

Recent Advances in Meth8/VŁ4, a Modal Model Checker with Universal Logic

Authors: Colin James III
Comments: 2 Pages. Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com; email: info@cec-services . com

We evaluate 712 artifacts in 3677 assertions to confirm 554 as tautology and 3123 as not (84.9%) in 1196 draft pages.
Category: Set Theory and Logic

[202] viXra:1911.0021 [pdf] replaced on 2019-11-02 04:12:14

Refutaton of Frege’s Inference Rule of Generalization

Authors: Colin James III
Comments: 1 Page. © Copyright 2016-2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Updated abstract at ersatz-systems.com.

We evaluate Frege’s inference rule of generalization which is not tautologous. This result forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[201] viXra:1910.0556 [pdf] replaced on 2019-10-28 01:53:49

Modus Inversus – if (Premise is False) Then (Conclusion is False)

Authors: Ilija Barukčić
Comments: 9 Pages.

Objective: When theorems or theories are falsified by a formal prove or by observations et cetera, authors respond many times by different and sometimes inappropriate counter-measures. Even if the pressure by which we are forced to believe in different theories although there are already predictively superior rivals to turn to may be very high, a clear scientific methodology should be able to help us to assure the demarcation between science and pseudoscience. Methods: Karl Popper’s (1902-1994) falsificationist methodology is one of the many approaches to the problem of the demarcation between scientific and non-scientific theories but relies as such too much only on modus tollens and is in fact purely one-eyed. Results: Modus inversus is illustrated in more detail in order to identify non-scientific claims as soon as possible and to help authors not to hide to long behind a lot of self-contradictory and sometimes highly abstract, even mathematical stuff. Conclusions: Modus inversus prevents us from accepting seemingly contradictory theorems or rules in science. Keywords: Science, non-science, modus inversus. E-Mail: Barukcic@t-online.de
Category: Set Theory and Logic

[200] viXra:1910.0443 [pdf] replaced on 2019-10-24 10:17:06

The Real Numbers are Denumerable

Authors: James Edwin Rock
Comments: 2 Pages.

We show Cantor’s diagonal argument has an invalid premise. We create a non-hierarchical Level Set Theory by setting 1/(Aleph Null) = 0. The real numbers have the same cardinality as the set of natural numbers, since the power set of the natural numbers has the same cardinality as the natural numbers. Using binary decimals, we define r, 0 ≤ r ≤ 1 as the ratio of ones to the total number of digits for decimals in the closed interval [0, 1]. We use these ratios to show that the real numbers are denumerable.
Category: Set Theory and Logic

[199] viXra:1910.0319 [pdf] replaced on 2019-12-02 06:49:37

Refutation of Relativized Boolean Algebra as Propositional Awareness Logic

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

From the definitions given, excepting three trivial tautologies as rendered, six equations are not tautologous, and one is contradictory. This refutes relativized Boolean algebras and what follows, namely: concrete relativized Boolean algebras; models of propositional logic; awareness frames and awareness models; modal RB’s; powerset MRBAs; and ultrafilter frames. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[198] viXra:1908.0489 [pdf] replaced on 2019-08-29 05:14:08

The Mathematical Expressions of Quranic Exegeses and the Mathematical Definition of the Quranic Correctness

Authors: Hiroki Tahara
Comments: 4 Pages.

I succeeded to give mathematical expressions to any correct Quranic Exegeses and define the Quranic correctness as the unique existence of Tahara I function. In a precise mathematical sense, the expressions and the definition are ill-defined however they might have meanings to prove the Quranic correctness.
Category: Set Theory and Logic

[197] viXra:1908.0489 [pdf] replaced on 2019-08-25 08:55:00

The Mathematical Expressions of Quranic Exegeses and the Mathematical Definition of the Quranic Correctness

Authors: Hiroki Tahara
Comments: 4 Pages.

I succeeded to give mathematical expressions to any correct Quranic exegeses and define the Quranic correctness as the unique existence of Tahara I function. In a precise mathematical sense, the expressions and the definition are ill-defined however they might have meanings to prove the Quranic correctness.
Category: Set Theory and Logic

[196] viXra:1908.0445 [pdf] replaced on 2019-08-23 19:04:22

Availability of Meth8/VŁ4 Demo for 2-Variables (P,q) with Unlimited Sequents

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

This is free on request from info@ersatz-systems.com. Please state name and organization to receive: Unrestricted m8_executable.exe; Instructions.txt with known anomalies; and Editable sample meth8_input_file.txt. The input file contains the shortest confirmation of McCune's proof of Huntington's equation: (~(~p+q)+~(~p+~q))=p. The input file contains the shortest refutation of paraconsistent logic: ((p>q)&(q>p))&((~p>~q)&(~q>~p)). The input file also contains the refutation for provability logic of the Gödel-Löb axiom GL as, "The necessity of choice, as always implying a choice, implies always a choice": #(#p>p)>#p.
Category: Set Theory and Logic

[195] viXra:1908.0363 [pdf] replaced on 2019-08-18 09:55:05

Scalability of Meth8/VL4

Authors: Colin James III
Comments: 1 Page.

We offer two more packages in Meth8 to process increasing numbers of propositional variables.
Category: Set Theory and Logic

[194] viXra:1908.0274 [pdf] replaced on 2019-08-14 10:40:22

Refutation of the Paradox of Hempel’s Raven

Authors: Colin James III
Comments: 2 Pages. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the hypothesis which is not tautologous and hence not a paradox. It forms a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[193] viXra:1908.0273 [pdf] replaced on 2019-08-14 10:41:21

Refutation of Differential Reasoning

Authors: Colin James III
Comments: 1 Page. Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We refute Hempel’s raven paradox elsewhere, hence refuting differential reasoning which forms a non tautologous fragment of the universal logic VL4.
Category: Set Theory and Logic

[192] viXra:1907.0053 [pdf] replaced on 2019-07-05 07:09:11

Refutation of the Curry-Howard Correspondence

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In the Curry-Howard correspondence, the identity and composition combinators are not tautologous. In fact, the examples result in equivalent truth table values. Further demonstrated is that the instances of Hilbert, lambda, and sequent fragments are also not tautologous with a recent paper rendered moot. These artifacts form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[191] viXra:1906.0407 [pdf] replaced on 2019-07-17 13:33:25

Visualizing the Distributions and Isosurfaces of Some Traditional and Non-Traditional Quaternion Fractal Sets

Authors: Shawn Halayka
Comments: 18 Pages.

After a concise introduction, the length, displacement, and magnitude distributions and isosurfaces related to some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[190] viXra:1906.0140 [pdf] replaced on 2019-07-13 09:21:21

The Logical Content of Inequalities

Authors: Ilija Barukčić
Comments: 20 pages. Copyright © 2019 by Ilija Baruk�?ić, Jever, Germany. All rights reserved. Published by: Ilija Baruk�?ić "The Interior Logic of Inequalities" IJMTT, 65 (7) (2019): 146-155. doi: 10.14445/22315373/IJMTT-V65I7P524

Objective. From a theoretical point of view, the demarcation between science and (fantastical) pseudoscience is in necessary for both practical and theoretical reasons. One specific nature of pseudoscience in relation to science and other categories of human reasoning is the resistance to the facts. Methods. Several methods are analyzed which may be of use to prevent that belief is masqueraded genuinely as scientific knowledge. Results. Modus ponens, modus tollens and modus conversus are reanalyzed. Modus sine, logically equivalent to modus ponens is developed. Modus inversus and modus juris are described in detail. Conclusions. In our striving for knowledge, there is still much more scientific work to be done on the demarcation between science and pseudoscience. Keywords: non strict inequality, quantum mechanics, science, pseudoscience
Category: Set Theory and Logic

[189] viXra:1905.0358 [pdf] replaced on 2019-09-12 06:06:15

Proposal of Replacing Classical Logic with Free Logic for Reasoning with Non-Referring Names in Ordinary Discourse

Authors: Bornali Paul
Comments: 9 Pages.

Reasoning carried out in ordinary language, can not avoid using non-referring names if occasion arises. Semantics of classical logic does not fit well for dealing with sentences with non-referring names of the language. The principle of bivalence does not allow any third truth-value, it does not allow truth-value gap also. The outcome is an ad hoc stipulation that no names should be referentless. The aim of this paper is to evaluate how far free logic with supervaluational semantics is appropriate for dealing with the problems of non-referring names used in sentences of ordinary language, at the cost of validity of some of the classical logical theses/ principles.
Category: Set Theory and Logic

[188] viXra:1905.0310 [pdf] replaced on 2019-05-19 22:47:28

Refutation of Tarski’s Geometric Axioms and Betweenness

Authors: Colin James III
Comments: 3 Pages.

Of 13 equations evaluated, five are tautologous and eight are non tautologous. This refutes Tarski’s geometric axioms and betweenness which form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[187] viXra:1904.0466 [pdf] replaced on 2019-04-25 08:14:09

Refutation of Strong Jump Inversion and Decidable Copy of a Saturated Model of DCF0

Authors: Colin James III
Comments: 1 Page.

From the paper’s abstract, the definition of strong jump inversion is not tautologous, hence strong jump inversion is refuted. A computable enumeration of the types realized in models of DCF0 is also refuted. The alleged fact that the saturated model of DCF0 has a decidable copy is denied. Therefore these conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[186] viXra:1904.0182 [pdf] replaced on 2019-04-22 06:07:09

Refutation of Fodor’s Causality Principle and Extension to the Class Fodor Principle

Authors: Colin James III
Comments: 2 Pages.

We evaluate Fodor’s principle in two versions we name weaker and stronger. Neither is tautologous. By extension, the class Fodor principle is not tautologous. Because of this, using Kelly-Morse set theory as a basis for denial is rendered moot. However, to rely on KM set theory to deny Fodor principles as a class principle is obviated by the refutation above. Therefore Fodor’s causality principles and extension are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[185] viXra:1903.0410 [pdf] replaced on 2019-03-23 07:50:39

Refutation of Coalgebraic Geometric Modal Logic

Authors: Colin James III
Comments: 2 Pages.

Two definitions equations from Eqs. 4.4.1.2 and 4.4.4.2 as rendered are not tautologous, hence denying the monotone functor on KHaus. What follows is that the use of coalgebra to manufacture a geometric modal logic is refuted. Therefore the conjecture is a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[184] viXra:1903.0206 [pdf] replaced on 2019-03-13 08:53:41

Universal logic VŁ4

Authors: Colin James III
Comments: 20 Pages.

This paper demonstrates why logic system VŁ4 is a universal logic composed of any refutation as a non-tautologous fragment. Recent advances are a definitive answer to criticism of logic Ł4, modal equations for lines and angles of the Square of Opposition, confirmation of the 24-syllogisms by updating Modus Cesare and Camestros, and proving that respective quantified and modal operators are equivalent. The parser Meth8 implements VŁ4 as the modal logic model checker Meth8/VŁ4. Over 430 assertions are tested in 2300 assertions with a refutation rate of 81%.
Category: Set Theory and Logic

[183] viXra:1903.0049 [pdf] replaced on 2019-05-31 19:43:12

Interval Sieve Algorithm - Creating a Countable Set of Real Numbers from a Closed Interval

Authors: Ron Ragusa
Comments: 6 Pages.

The interval sieve algorithm partitions a closed interval of real numbers [ri, rj] where ri < rj to create a complete list, L, of numbers in the interval. We will prove that the list L is complete, and derive the bijective function f : ℕ → [r1, r2].
Category: Set Theory and Logic

[182] viXra:1903.0049 [pdf] replaced on 2019-05-23 14:58:21

Interval Sieve Algorithm - Creating a Countable Set of Real Numbers from a Closed Interval

Authors: Ron Ragusa
Comments: 6 Pages.

The interval sieve algorithm partitions a closed interval of real numbers [ri, rj] where ri < rj to create a complete list, L, of numbers in the interval. We will prove that the list L is complete, and lastly derive the bijective function f : ℕ ↔ [r1, r2].
Category: Set Theory and Logic

[181] viXra:1903.0049 [pdf] replaced on 2019-05-16 22:30:10

Interval Sieve Algorithm - Creating a Countable Set of Real Numbers from a Closed Interval

Authors: Ron Ragusa
Comments: 6 Pages.

The interval sieve algorithm partitions a closed interval of real numbers [ri, rj] where ri < rj to create a complete list, L, of numbers in the interval. We will prove that the list L is complete, and lastly derive the bijective function f : ℕ ↔ [r1, r2].
Category: Set Theory and Logic

[180] viXra:1903.0049 [pdf] replaced on 2019-03-25 22:29:33

Interval Sieve Algorithm - Creating a Countable Set of Real Numbers from a Closed Interval

Authors: Ron Ragusa
Comments: 5 Pages.

The Interval Sieve Algorithm is a method for generating a list of real numbers on any closed interval of real numbers [ri, rj] where ri < rj. The purpose of this paper is to: 1.Present the algorithm and demonstrate the construction of the list L, 2.Prove that the list L is complete, 3.Derive the bijective function f : ℕ ↔ [r1, r2].
Category: Set Theory and Logic

[179] viXra:1903.0049 [pdf] replaced on 2019-03-08 06:41:25

Interval Sieve Algorithm - Creating a Countable Set of Real Numbers from a Closed Interval

Authors: Ron Ragusa
Comments: 6 Pages.

The Interval Sieve Algorithm is a method for generating a list of real numbers on any closed interval [ri, rj] where ri < rj, which can then be defined as the domain of the function f(x) = C. The purpose of this paper is to delineate the steps of the algorithm and show how it will generate a countable list from which the domain for the function f(x) = C can be defined. Having constructed the list we will prove that the list is complete, that it contains all the numbers in the interval [ri, rj]. Lastly we will demonstrate a restricted proof of the Continuum Hypothesis.
Category: Set Theory and Logic

[178] viXra:1903.0049 [pdf] replaced on 2019-03-06 16:13:39

Interval Sieve Algorithm - Creating a Countable Set of Real Numbers from a Closed Interval

Authors: Ron Ragusa
Comments: 6 Pages.

The Interval Sieve Algorithm is a method for generating a list of real numbers on any closed interval [ri, rj] where ri < rj, which can then be defined as the domain of the function f(x) = C. The purpose of this paper is to delineate the steps of the algorithm and show how it will generate a countable list from which the domain for the function f(x) = C can be defined. Having constructed the list we will prove that the list is complete, that it contains all the numbers in the interval [ri, rj]. Lastly we will demonstrate a restricted proof of the Continuum Hypothesis.
Category: Set Theory and Logic

[177] viXra:1903.0049 [pdf] replaced on 2019-03-05 22:33:09

Interval Sieve Algorithm - Creating a Countable Set of Real Numbers from a Closed Interval

Authors: Ron Ragusa
Comments: 7 Pages.

The Interval Sieve Algorithm is a method for generating a list of real numbers on any closed interval [ri, rj] where ri < rj, which can then be defined as the domain of the function f(x) = C. The purpose of this paper is to delineate the steps of the algorithm and show how it will generate a countable list from which the domain for the function f(x) = C can be defined. Having constructed the list we will prove that the list is complete, that it contains all the numbers in the interval [ri, rj].
Category: Set Theory and Logic

[176] viXra:1902.0451 [pdf] replaced on 2019-02-27 10:15:14

Refutation of Definable Operators on Stable Set Lattices

Authors: Colin James III
Comments: 1 Page.

We evaluate the definitions for the modal operators on stable set lattices. The operators are not respective negations and hence refute the definitions.
Category: Set Theory and Logic

[175] viXra:1902.0402 [pdf] replaced on 2019-02-24 06:12:22

Refutation of Relativization by Structural Induction in Weighted First Order Logic

Authors: Colin James III
Comments: 1 Page.

We evaluate a formula of relativization as defined by structural induction which is not tautologous. Its use in weighted first order logic is refuted.
Category: Set Theory and Logic

[174] viXra:1902.0397 [pdf] replaced on 2019-04-23 21:09:03

Refutation of the Banach-Tarski Paradox

Authors: Colin James III
Comments: 1 Page.

We use the Meth8/VŁ4 modal logic model checker to evaluate over 450 artifacts in 2400 assertions to refute 1950 as non tautologous (81%). This is the table of contents for over 700 pages; current document available on request.
Category: Set Theory and Logic

[173] viXra:1902.0281 [pdf] replaced on 2019-02-18 20:11:18

Refutation of Peirce's Abduction and Induction, and Confirmation of Deduction

Authors: Colin James III
Comments: 2 Pages.

We evaluate definitions of C.S. Peirce for abduction, induction and deduction: all are inversions of the same sentence. However, when the connectives are changed to implication, abduction and induction are not tautologous, leaving deduction as the only form of tautologous inference in logic.
Category: Set Theory and Logic

[172] viXra:1902.0249 [pdf] replaced on 2019-02-15 08:01:15

Meth8/VŁ4 on One and Three in Arithmetic

Authors: Colin James III
Comments: 2 Pages.

We evaluate arithmetic using 0 and 3 as binary 00 and 11.. Arithmetic holds in nine theorems. For division by zero, the result is Not(0 and 3).
Category: Set Theory and Logic

[171] viXra:1902.0197 [pdf] replaced on 2019-02-11 21:48:30

Refutation of the Method of Forcing

Authors: Colin James III
Comments: 2 Pages.

We evaluate two examples of paradoxes to be resolved by the method of forcing. The first is not tautologous, and the second is a contradiction so not proved as a paradox. This means the forcing method is refuted because it cannot coerce the two examples into abstract proofs and with the ultimate goal to produce larger truth values. What follows is that the forcing method is better suited for paraconsistent logics which are non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[170] viXra:1902.0189 [pdf] replaced on 2019-02-11 10:01:58

The Philosophy of Mathematics

Authors: Hugh Wang
Comments: 2 Pages.

What is mathematics? Why does it exist? Is it consistent? Is it complete? We answer these questions as well as resolve Russell’s Paradox and debunk Godel’s Incompleteness Theorem.
Category: Set Theory and Logic

[169] viXra:1902.0167 [pdf] replaced on 2019-02-10 05:38:45

Denial of the Refutation of Coherence in Modal Logic

Authors: Colin James III
Comments: 1 Page.

We evaluate the refutation of coherence in modal logic as based on weakly transitive logics using a ternary term to admit finite chains. The term is not tautologous, thereby denying the refutation. What follows is that K, KT, K4, and S4 (and S5) are tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[168] viXra:1902.0141 [pdf] replaced on 2019-02-08 06:38:43

Denial of Logic System PŁ4

Authors: Colin James III
Comments: 2 Pages.

We evaluate the logic system PŁ4 as refuting and replacing VŁ4. Eight modal theses and two axioms are not tautologous and contrary to those of PŁ4. This denies that PŁ4 refutes VŁ4, refutes PŁ4, and justifies VŁ4 as containing the non bivalent fragment named PŁ4.
Category: Set Theory and Logic

[167] viXra:1902.0140 [pdf] replaced on 2019-02-08 05:19:56

Two VŁ4 Theorems not in S5

Authors: Colin James III
Comments: 2 Pages.

We evaluate two equations not found as theorems in S5; both are theorems in VŁ4.
Category: Set Theory and Logic

[166] viXra:1902.0065 [pdf] replaced on 2019-02-07 20:08:21

Construction of Sets and Axiomatic System

Authors: Baekhyun Lee
Comments: 15 Pages. Language: Korean

This paper is written to show that concept of sets can be constructed with a few axioms, and also the axioms can be treated as a set.
Category: Set Theory and Logic

[165] viXra:1902.0065 [pdf] replaced on 2019-02-05 01:43:09

Construction of Sets and Axiomatic System

Authors: Baekhyun Lee
Comments: 15 Pages. Language: Korean

This paper is written to show that concept of sets can be constructed with a few axioms, and also the axioms can be treated as a set.
Category: Set Theory and Logic

[164] viXra:1902.0037 [pdf] replaced on 2019-02-03 06:27:03

Refutation of Confluence in Rewrite Systems

Authors: Colin James III
Comments: 2 Pages.

We evaluate confluence in two definitions and one theorem, none which is tautologous. This refutes the approach of confluence as a central property of rewrite systems.
Category: Set Theory and Logic

[163] viXra:1901.0342 [pdf] replaced on 2019-01-24 20:27:58

Refutation of Descriptive Unions in Descriptively Near Sets

Authors: Colin James III
Comments: 3 Pages.

We evaluate an intersection operator named descriptive union for descriptively near sets. From two sources the definition of the operator is not tautologous. A proof of seven properties derived from the second definition contains two trivial tautologies with the rest as not tautologous. This refutes the descriptive intersection operator and descriptively near sets on which it is based. This also casts doubt on the bevy of derived math and physics papers so spawned at arxiv, researchgate, and vixra.
Category: Set Theory and Logic

[162] viXra:1901.0342 [pdf] replaced on 2019-01-23 18:12:00

Refutation of Descriptive Unions in Descriptively Near Sets

Authors: Colin James III
Comments: 2 Pages.

We evaluate an intersection operator named descriptive union for descriptively near sets. A proof of seven properties contains two trivial tautologies and the rest as not tautologous. This refutes the descriptive intersection operator and descriptively near sets on which it is based. This also casts doubt on the derived slug of math and physics papers as spawned at arxiv, researchgate, and vixra.
Category: Set Theory and Logic

[161] viXra:1901.0192 [pdf] replaced on 2019-01-14 18:02:34

Refutation of Hexagons of Opposition for Statistical Modalities

Authors: Colin James III
Comments: 3 Pages.

Abstract: We evaluate pragmatic hypotheses in the evolution of science as based on probabilistic squares and hexagons of opposition under coherence. Neither conjecture is tautologous, and hence both are refuted.
Category: Set Theory and Logic

[160] viXra:1901.0069 [pdf] replaced on 2019-01-28 02:47:12

Monoidal Structures of Logic

Authors: Antoine Balan
Comments: 2 pages, written in french

The tables of truth are relations of a monoid M.
Category: Set Theory and Logic

[159] viXra:1812.0423 [pdf] replaced on 2019-10-08 16:31:29

The Curvature and Dimension of a Closed Surface

Authors: Shawn Halayka
Comments: 7 Pages.

The curvature of a closed surface can lead to fractional dimension. In this paper, the properties of the $2$-sphere surface of a three-dimensional ball and the $2.x$-dimensional surface of a three-dimensional fractal set are considered. Tessellation is used to approximate each surface, primarily because the $2.x$-dimensional surface of a three-dimensional fractal set is otherwise non-differentiable (having no well-defined surface normals).
Category: Set Theory and Logic

[158] viXra:1812.0423 [pdf] replaced on 2019-09-26 22:35:40

The Curvature and Dimension of a Closed Surface

Authors: Shawn Halayka
Comments: 5 Pages.

The curvature of a closed surface can lead to fractional dimension. In this paper, the properties of the $2$-sphere surface of a three-dimensional ball and the $2.x$-dimensional surface ($2.x$-surface) of a three-dimensional fractal set are considered. Tessellation is used to approximate each surface, primarily because the $2.x$-surface of a three-dimensional fractal set is otherwise non-differentiable (having no well-defined surface normals).
Category: Set Theory and Logic

[157] viXra:1812.0423 [pdf] replaced on 2019-09-21 20:55:02

The Curvature and Dimension of a Fractal Surface

Authors: Shawn Halayka
Comments: 8 Pages.

The curvature of a surface can lead to fractional dimension. In this paper, the properties of the $2$-sphere surface of a three-dimensional ball and the $2.x$-surface of a three-dimensional fractal set are considered. Tessellation is used to approximate each surface, primarily because the 2.$x$-surface of a three-dimensional fractal set is otherwise non-differentiable (having no well-defined surface normal).
Category: Set Theory and Logic

[156] viXra:1812.0423 [pdf] replaced on 2019-09-19 16:36:41

The Curvature and Dimension of Non-Differentiable Surfaces

Authors: Shawn Halayka
Comments: 6 Pages.

The curvature of a surface can lead to fractional dimension. In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered. Tessellation is used to approximate each surface, primarily because the 2.x-surface of a 3D fractal set is otherwise non-differentiable.
Category: Set Theory and Logic

[155] viXra:1812.0423 [pdf] replaced on 2019-01-06 16:35:48

The Curvature and Dimension of Non-Differentiable Surfaces

Authors: Shawn Halayka
Comments: 5 Pages.

The curvature of a surface can lead to fractional dimension. In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered.
Category: Set Theory and Logic

[154] viXra:1812.0423 [pdf] replaced on 2018-12-27 14:44:30

The Curvature and Dimension of Surfaces

Authors: Shawn Halayka
Comments: 5 Pages.

The curvature of a surface can lead to fractional dimension. In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered.
Category: Set Theory and Logic

[153] viXra:1812.0423 [pdf] replaced on 2018-12-26 15:49:33

The Curvature and Dimension of Surfaces

Authors: Shawn Halayka
Comments: 4 Pages.

The curvature of surfaces can lead to fractional dimension. In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered.
Category: Set Theory and Logic

[152] viXra:1812.0275 [pdf] replaced on 2018-12-17 12:21:39

Confirmation of the Łukasiewicz Square of Opposition Via Logic VŁ4

Authors: Colin James III
Comments: 4 Pages. © Copyright 2016-2019 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the existential import of the Revised Modern Square of Opposition. We confirm that the Łukasiewicz syllogistic was intended to apply to all terms. What follows is that Aristotle was mistaken in his mapping of vertices, which we correct and show fidelity to Aristotle's intentions. We also evaluate the Cube of Opposition of Seuren. Two final claims are not tautologous, hence refuting the Cube, which also contradict criticism of Seuren that was not based on those claims.
Category: Set Theory and Logic

[151] viXra:1812.0209 [pdf] replaced on 2019-02-03 06:50:24

Refutation of the Hahn-Banach Theorem

Authors: Colin James III
Comments: 2 Pages.

We evaluate the Hahn-Banach theorem. Without or with the universal quantifiers, the equations are not tautologous. This refutes the Hahn-Banach theorem.
Category: Set Theory and Logic

[150] viXra:1812.0168 [pdf] replaced on 2019-03-28 17:38:13

Refutation of the Constructive Craig Interpolation Theorem by Feferman

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the Craig constructive interpolation theorem, find a mistake in a Craig lemma as rendered by Feferman, and refute the theorem as not constructive.
Category: Set Theory and Logic

[149] viXra:1812.0047 [pdf] replaced on 2018-12-04 11:12:13

Refutation of the Lonely Runner Conjecture with Three Runners

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the conjecture of the lonely runner with three runners. We do not assume a runner may be stationary as a no-go contestant. The result is that the conjecture diverges from tautology by one logical value and hence is refuted. We then assume a runner can be stationary with result of the same truth table also to refute the conjecture.
Category: Set Theory and Logic

[148] viXra:1812.0016 [pdf] replaced on 2018-12-03 12:51:54

Refutation of the Frauchiger-Renner Thought Paradox as a Quantum Model

Authors: Colin James III
Comments: 4 Pages.

We use modal logic to evaluate a quantum model of the Frauchiger-Renner thought experiment as not a contradiction (paradox) and not a tautology (theorem). The example misapplies the Born rule which we refute elsewhere.
Category: Set Theory and Logic

[147] viXra:1811.0439 [pdf] replaced on 2018-12-24 09:07:26

Remark on Seven Applications of Neutrosophic Logic: in Cultural Psychology, Economics Theorizing, Conflict Resolution, Philosophy of Science, Etc.

Authors: Victor Christianto, Florentin Smarandache
Comments: 12 Pages. This paper has been submitted to J. MATHEMATICS. Your comments are welcome

In this short communication, we review seven applications of NFL which we have explored in a number of papers: 1) Background: The purpose of this study is to review on how Neutrosophic Logic can be found useful in a number of diverse areas of interest; 2) Methods: we use logical analysis based on NL; 3) Results: Some fields of study may be found elevated after analyzed by NL theory; and 4) Conclusions: We can expect NL theory can be applied in many areas of research too, both in applied mathematics, economics, and also physics. Hopefully the readers will find a continuing line of thoughts in our research in the last few years
Category: Set Theory and Logic

[146] viXra:1807.0418 [pdf] replaced on 2018-09-24 22:11:51

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 8 Pages.

The escape paths of the points in some quaternion fractal sets are visualized.
Category: Set Theory and Logic

[145] viXra:1807.0418 [pdf] replaced on 2018-09-20 15:47:34

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 8 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL 1.x. C++ source code is provided.
Category: Set Theory and Logic

[144] viXra:1807.0418 [pdf] replaced on 2018-09-18 12:32:44

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 8 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[143] viXra:1807.0418 [pdf] replaced on 2018-09-17 17:04:49

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[142] viXra:1807.0418 [pdf] replaced on 2018-08-01 13:29:49

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 4 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[141] viXra:1807.0418 [pdf] replaced on 2018-07-30 09:50:33

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[140] viXra:1807.0418 [pdf] replaced on 2018-07-26 16:18:11

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 7 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[139] viXra:1806.0030 [pdf] replaced on 2018-06-27 22:04:44

The Function f(x) = C and the Continuum Hypothesis

Authors: Ron Ragusa
Comments: 14 Pages. email: ron.ragusa@gmail.com

Part 1 examines whether or not an analysis of the behavior of the function f(x) = C, where C is any constant, on the interval (a, b) where a and b are real numbers and a < b, will provide a method of proving the truth or falsity of the Continuum Hypothesis (CH). The argument will be presented in three theorems and one corollary. The first theorem proves, by construction, the countability of the domain d of f(x) = C on the interval (a, b) where a and b are real numbers. The second theorem proves, by substitution, that the set of natural numbers ℕ has the same cardinality as the subset S of real numbers on the given interval. The corollary extends the proof of theorem 2 to show that ℕ and ℝ are of the same cardinality. The third theorem proves, by logical inference, that the CH is true. Part 2 is a demonstration of how the set of natural numbers ℕ can be put into a one to one correspondence with the power set of natural numbers, P(ℕ). From this I will derive the bijective function f : ℕ → P(ℕ). Lastly, I’ll propose a conjecture asserting that f(x) = C can be employed to construct a one to one correspondence between the natural numbers and any infinite set that can be cast as the domain of the function. Appendix A extends the methodology for creating a bijection between infinite sets to the function f(x) = x2 using random real numbers from the domain of the function as input to f(x) = x2 in order to show how the constructed array would appear in practical application.
Category: Set Theory and Logic

[138] viXra:1806.0030 [pdf] replaced on 2018-06-04 08:33:05

The Function f(x) = C and the Continuum Hypothesis

Authors: Ron Ragusa
Comments: 5 Pages.

This paper examines whether or not an analysis of the behavior of the continuous function f(x) = C, where C is any constant, on the interval (a, b) where a and b are real numbers and a < b, will provide a method of proving the truth or falsity of the CH. The argument will be presented in three theorems and one corollary. The first theorem proves, by construction, the countability of the domain d of f(x) = C on the interval (a, b) where a and b are real numbers. The second theorem proves, by substitution, that the set of natural numbers N has the same cardinality as the subset of real numbers S on the given interval. The corollary extends the proof of theorem 2 to show that N and R are of the same cardinality. The third theorem proves, by logical inference, that the CH is true.
Category: Set Theory and Logic

[137] viXra:1805.0543 [pdf] replaced on 2019-07-19 21:59:26

Refutation of the Modern, General, and Strong Löwenheim–Skolem Theorem

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the equation for the modern, general, and strong Löwenheim–Skolem theorem. It is not tautologous, hence refuting the upward and downward parts. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[136] viXra:1805.0302 [pdf] replaced on 2018-05-16 23:04:06

Refutation of Gettier Problem of Justified True/false Belief © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

We find justified true belief is not a theorem. We find justified false belief is also not a theorem. This means the Gettier problem as the superset of the justified belief arguments is refuted as a problem and resolved as a non-problem.
Category: Set Theory and Logic

[135] viXra:1805.0138 [pdf] replaced on 2018-05-09 04:48:16

Refutation of the Paradox of Moses Maimonides © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Does God know or does He not know that a certain individual will be good or bad? If thou sayest 'He knows', then it necessarily follows that the man is compelled to act as God knew beforehand he would to act, otherwise God's knowledge would be imperfect. The question is tautologous. The additional sentence with the first is not tautologous and not contradictory. Therefore the paradox of Maimonides is refuted as a paradox.
Category: Set Theory and Logic

[134] viXra:1805.0124 [pdf] replaced on 2018-05-06 14:30:48

Refutation of the Prisoner Paradox© Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

The three states of the prisoner paradox are not contradictory. This means the prisoner paradox is not a paradox.
Category: Set Theory and Logic

[133] viXra:1804.0174 [pdf] replaced on 2018-05-04 01:07:50

The Universe Equation

Authors: Seamus McCelt
Comments: 2 Pages.

If you claim there are particles: there would actually have to be particles.
And that would mean there are about 18 different microscopic things that work flawlessly together -- just like clockwork to make even just one basic atom "gear" set.
If you have larger sized atoms: it would be like throwing more and more gear sets into the clockwork -- but that is ok because no matter what you throw in -- it will still work just fine.
How can an infinity of 18 different things (infinity times 18 different things) just happen to be here, know how work together as a group and also successfully work together as a group(s)?
How is that possible? It isn't...

Stuff cannot be made from what they call "particles."

If there are particles; this is equation of the universe:
Universe = Infinity × {a,b,c,d,f,g,h,j,k,l,m,o,p,q,t,w,x,y,z}
Category: Set Theory and Logic

[132] viXra:1803.0318 [pdf] replaced on 2018-03-21 06:51:30

Refutation of Abductive Reasoning © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

Abductive logic of C.S. Peirce is refuted as not tautologous.
Category: Set Theory and Logic

[131] viXra:1803.0180 [pdf] replaced on 2019-07-29 11:01:31

Refutation of the Euathlus Paradox: Neither Pay © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. . © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Regardless of who wins the lawsuit of Portagoras, Euathlus does not pay. Hence the Euathlus paradox is refuted and resolved by default in favor of Euathlus.
Category: Set Theory and Logic

[130] viXra:1803.0094 [pdf] replaced on 2018-03-07 17:10:19

Refutation of Cantor's Original Continuum Hypothesis Via Injection and Binary Trees © Copyright 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved.

This is the briefest known such refutation of Cantor's continuum conjecture.
Category: Set Theory and Logic

[129] viXra:1712.0403 [pdf] replaced on 2019-05-09 16:59:48

There is No Standard Model of ZFC and ZFC_2

Authors: Jaykov Foukzon
Comments: 73 Pages.

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st),(ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k)
Category: Set Theory and Logic

[128] viXra:1712.0403 [pdf] replaced on 2017-12-23 02:39:19

There is no Standard Model of ZFC

Authors: Jaykov Foukzon
Comments: 14 Pages. Journal of Global Research in Mathematical Archives (JGRMA)Vol 5, No 1 (2018): January-2018

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st),(ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k),[10],11].
Category: Set Theory and Logic

[127] viXra:1712.0403 [pdf] replaced on 2017-12-15 05:52:44

There is no Standard Model of ZFC

Authors: Jaykov Foukzon
Comments: 13 Pages.

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st),(ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k),[10],[11].
Category: Set Theory and Logic

[126] viXra:1712.0139 [pdf] replaced on 2018-04-03 05:31:20

A Proof of the Falsity of the Axiom of Choice.

Authors: Johan Noldus
Comments: 1 Page.

We show that the axiom of choice is false.
Category: Set Theory and Logic

[125] viXra:1712.0139 [pdf] replaced on 2017-12-22 12:51:03

A Proof of the Falsity of the Axiom of Choice.

Authors: Johan Noldus
Comments: 1 Page.

We show that the axiom of choice is false.
Category: Set Theory and Logic

[124] viXra:1711.0425 [pdf] replaced on 2017-11-26 08:56:58

Refutation of Realizability Semantics for QML © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2017 by Colin James III All rights reserved.

We found that the following are not tautologous: Epistemic Church's Thesis; EZF induction schema; and Scedrov's modal foundation. We did not test subsequent axioms. What follows is that Flagg's construction, Goodman's intensional set theory, and epistemic logic are suspicious.
Category: Set Theory and Logic

[123] viXra:1711.0113 [pdf] replaced on 2018-12-17 08:46:05

Tautology Problem and Two Dimensional Formulas

Authors: Deniz Uyar
Comments: 39 Pages.

Finding whether a boolean formula is a tautology or not in a feasible time is an important problem of computer science. Many algorithms have been developed to solve this problem but none of them is a polynomial time algorithm. Our aim is to develop an algorithm that achieve this in polynomial time. In this article, we convert boolean functions to some graph forms in polynomial time. They are called two dimensional formulas and similar to AND-OR graphs except arches on them are bidirectional. Then these graphs are investigated to find properties which can be used to differentiate tautological formulas from non tautological ones.
Category: Set Theory and Logic

[122] viXra:1708.0156 [pdf] replaced on 2018-03-14 06:03:39

Some Critical Notes on the Cantor Diagonal Argument

Authors: Philip Molyneux
Comments: 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in set theory to draw a distinction between the cardinality of the natural numbers and that of the real numbers. In the absence of a verified English translation of the original 1891 Cantor paper from which it is said to be derived, the CDA is discussed here using a consensus from the forms found in a range of published sources (from "popular" to "professional"). Some general comments are made on these sources. The discussion then focusses on the CDA as applied to the correspondence between the set of the natural numbers, and the set of real numbers in the open range (0,1) in their expansion from decimal digits (0.123… etc.). Four points critical of the CDA are raised: (1) The conventional presentation of the CDA forms a putative new real number (X) from the "diagonal" of the chosen list of real numbers and which is therefore not on this initial list; however, it omits to consider that it may indeed be on the later part of the list, which is never exhausted however far the "diagonal" list is extended. (2) This aspect, combined with the fact that X is still composed of decimal digits, that is, it is a real number as defined, indicates that it must be on the later part of the list, that is, it is not a "new" number at all. (3) The conventional application of the CDA apparently leads to one putative "new" real number (X); however, the logical extension of this in its "exhaustive" application, that is, by using all possible different methods of alteration of the decimal digits on the "diagonal", and by reordering the list in all possible ways, leads to a list of putative "new" real numbers that become orders of magnitude longer than the chosen "diagonal" list. (4) The CDA is apparently considered to be a method that is applicable generally; however, testing this applicability with the natural numbers themselves leads to this contradiction. Following on from this, it is found that it indeed is possible to set up a one-to-one correspondence between the natural numbers and the real numbers in (0,1), that is, ! ⇔ "; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension of the natural number is intended to be a mirror image of the left hand extension of the real number. This may be extended to the general case of real numbers - that is, not limited to the range (0,1) - by intercalation of the digit sequence of its decimal fraction part into the sequence of the natural number part, giving the one-to-one-correspondence: … A3 a3 A2 a2 A1 a1 ⇔ ... A3 A2 A1. a1 a2 a3 … Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998 critical review of "hopeless papers" dealing with the CDA; this form is also examined from the same viewpoints, and to the same conclusions. Finally, some comments are made on the concept of "infinity", pointing out that to consider this as an entity is a category error, since it simply represents an absence, that is, the absence of a termination to a process.
Category: Set Theory and Logic

[121] viXra:1708.0156 [pdf] replaced on 2017-08-24 10:23:39

Some Critical Notes on the Cantor Diagonal Argument

Authors: Philip Molyneux
Comments: 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in set theory to draw a distinction between the cardinality of the natural numbers and that of the real numbers. In the absence of a verified English translation of the original 1891 Cantor paper from which it is said to be derived, the CDA is discussed here using a consensus from the forms found in a range of published sources (from "popular" to "professional"). Some general comments are made on these sources. The discussion then focusses on the CDA as applied to the correspondence between the set of the natural numbers, and the set of real numbers in the open range (0,1) in their expansion from decimal digits (0.123… etc.). Four points critical of the CDA are raised: (1) The conventional presentation of the CDA forms a putative new real number (X) from the "diagonal" of the chosen list of real numbers and which is therefore not on this initial list; however, it omits to consider that it may indeed be on the later part of the list, which is never exhausted however far the "diagonal" list is extended. (2) This aspect, combined with the fact that X is still composed of decimal digits, that is, it is a real number as defined, indicates that it must be on the later part of the list, that is, it is not a "new" number at all. (3) The conventional application of the CDA leads to one putative "new" real number (X); however, the logical extension of this in its "exhaustive" application, that is, by using all possible different methods of alteration of the decimal digits on the "diagonal", and by reordering the list in all possible ways, leads to a list of putative "new" real numbers that become orders of magnitude longer than the chosen "diagonal" list. (4) The CDA is apparently considered to be a method that is applicable generally; however, testing this applicability with the natural numbers themselves leads to a contradiction. Following on from this, it is found that it indeed is possible to set up a one-to-one correspondence between the natural numbers and the real numbers in (0,1), that is, ℕ ⇔ ℝ; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension of the natural number is intended to be a mirror image of the left hand extension of the real number. It is also shown how this may be extended to real numbers outside the range (0,1). Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998 critical review of "hopeless papers" dealing with the CDA; this form is also examined from the same viewpoints, and to the same conclusions. Finally, some comments are made on the concept of "infinity", pointing out that to consider this as an entity is a category error, since it simply represents an absence, that is, the absence of a termination to a process.
Category: Set Theory and Logic

[120] viXra:1708.0156 [pdf] replaced on 2017-08-22 04:28:33

Some Critical Notes on the Cantor Diagonal Argument

Authors: Philip Molyneux
Comments: 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in set theory to draw a distinction between the cardinality of the natural numbers and that of the real numbers. In the absence of a verified English translation of the original 1891 Cantor paper from which it is said to be derived, the CDA is discussed here using a consensus from the forms found in a range of published sources (from "popular" to "professional"). Some general comments are made on these sources. The discussion then focusses on the CDA as applied to the correspondence between the set of the natural numbers, and the set of real numbers in the open range (0,1) in their expansion from decimal digits (0.123… etc.). Four points critical of the CDA are raised: (1) The conventional presentation of the CDA forms a putative new real number (X) from the "diagonal" of the chosen list of real numbers and which is therefore not on this initial list; however, it omits to consider that it may indeed be on the later part of the list, which is never exhausted however far the "diagonal" list is extended. (2) This aspect, combined with the fact that X is still composed of decimal digits, that is, it is a real number as defined, indicates that it must be on the later part of the list, that is, it is not a "new" number at all. (3) The conventional application of the CDA leads to one putative "new" real number (X); however, the logical extension of this in its "exhaustive" application, that is, by using all possible different methods of alteration of the decimal digits on the "diagonal", and by reordering the list in all possible ways, leads to a list of putative "new" real numbers that become orders of magnitude longer than the chosen "diagonal" list. (4) The CDA is apparently considered to be a method that is applicable generally; however, testing this applicability with the natural numbers themselves leads to a contradiction. Following on from this, it is found that it indeed is possible to set up a one-to-one correspondence between the natural numbers and the real numbers in (0,1), that is, N ⇔ R; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension of the natural number is intended to be a mirror image of the left hand extension of the real number. It is also shown how this may be extended to real numbers outside the range (0,1). Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998 critical review of "hopeless papers" dealing with the CDA; this form is also examined from the same viewpoints, and to the same conclusions. Finally, some comments are made on the concept of "infinity", pointing out that to consider this as an entity is a category error, since it simply represents an absence, that is, the absence of a termination to a process.
Category: Set Theory and Logic

[119] viXra:1705.0173 [pdf] replaced on 2017-05-12 13:18:20

Dialectical Logic – Negation Of Classical Logic

Authors: Ilija Barukčić
Comments: 24 pages. Copyright © 2017 by Ilija Barukčić, Jever, Germany. Published by:

The division of zero by zero turns out to be a long lasting and not ending puzzle in mathematics and physics. An end of this long discussion is not in sight. In particular zero divided by zero is treated as indeterminate thus that a result cannot be found out. It is the purpose of this publication to solve the problem of the division of zero by zero while relying on the general validity of classical logic. According to classical logic, zero divided by zero is one.
Category: Set Theory and Logic

[118] viXra:1704.0115 [pdf] replaced on 2017-05-16 21:36:12

The Simple Infinite Set

Authors: Ken Seton
Comments: 9 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (say critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we identify the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite . Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be. Set theory found itself in some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.
Category: Set Theory and Logic

[117] viXra:1704.0115 [pdf] replaced on 2017-05-11 01:29:26

The Simple Infinite Set

Authors: Ken Seton
Comments: 9 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (say critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we identify the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite . Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be. Set theory found itself in some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.
Category: Set Theory and Logic

[116] viXra:1703.0032 [pdf] replaced on 2017-03-06 04:56:10

The Union is not the Limit.

Authors: Wolfgang Mückenheim
Comments: 5 Pages.

Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of sets differ.
Category: Set Theory and Logic

[115] viXra:1702.0293 [pdf] replaced on 2017-02-26 04:27:32

Not Enumerating All Positive Rational Numbers

Authors: W. Mückenheim
Comments: 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.
Category: Set Theory and Logic

[114] viXra:1702.0293 [pdf] replaced on 2017-02-24 05:08:54

Not Enumerating All Positive Rational Numbers

Authors: W. Mückenheim
Comments: 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.
Category: Set Theory and Logic

[113] viXra:1701.0563 [pdf] replaced on 2017-01-31 11:38:56

بیانیه ی یگانه و انجمن

Authors: امیر دلجو
Comments: 12 Pages.

این سند یک منشور است برای بیان ماهیت ریاضیات و نظریه ی اعداد یا حساب. در اینجا من الگویی را تعریف کرده ام که نشان می دهد، خودآگاهی یک وجود واحده بسیط و همه جاحاضر بوده و هستی به مثابه یک کلیّت، یک تجلّی گرافیکی ست که در جهت افشای این خودآگاهی عارض شده و من هرمس وار، شهود خود بر یگانگی و آفرینش را برای انتقال به انسان های کنجکاو ساده سازی و تحریر کرده ام. از آنجا که تبیین روش شناختی این منشور مستلزم هزاران صفحه است، در اینجا تنها عبارات و معادلات منتج شده ی نهایی اعلان می شود.
Category: Set Theory and Logic