# Set Theory and Logic

## 1806 Submissions

 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

 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

 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

 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

 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

 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

 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

 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