Set Theory and Logic

1902 Submissions

[40] 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

[39] 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

[38] 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

[37] 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

[36] 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

[35] 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

[34] 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

[33] 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

[32] 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

[31] 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

[30] 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

[29] 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

[28] 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

[27] 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

[26] 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

[25] 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

[24] 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

[23] 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

[22] 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

[21] 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

[20] 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

[19] 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

[18] 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

[17] 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

[16] 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

[15] 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

[14] 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

[13] 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

[12] 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

[11] 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

[10] 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

[9] 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

[8] 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

[7] 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

[6] 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

[5] 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

[4] 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

[3] 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

[2] 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

[1] 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