Set Theory and Logic

1910 Submissions

[26] 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

[25] 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

[24] 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

[23] 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

[22] 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

[21] 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

[20] 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

[19] 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

[18] 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

[17] 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

[16] 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

[15] 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

[14] 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

[13] 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

[12] 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

[11] 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

[10] 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

[9] 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

[8] 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

[7] 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

[6] 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

[5] 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

[4] 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

[3] 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

[2] 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

[1] 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