[9] **viXra:1806.0338 [pdf]**
*submitted on 2018-06-22 06:37:14*

**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

[8] **viXra:1806.0228 [pdf]**
*submitted on 2018-06-18 19:36:58*

**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.

We evaluate the following modal logic systems: (2, M); (3); (4); (B); (D);(E); (K); (T); and (W). Those diverging from tautology are: (3) as truthity; and (W) as not tautology or truthity. Systems K+W is Gödel's logic (GL), refuted in another paper here.

**Category:** Set Theory and Logic

[7] **viXra:1806.0213 [pdf]**
*submitted on 2018-06-19 14:16:30*

**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

[6] **viXra:1806.0162 [pdf]**
*submitted on 2018-06-12 23:21:45*

**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

[5] **viXra:1806.0113 [pdf]**
*submitted on 2018-06-09 22:49:48*

**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

[4] **viXra:1806.0109 [pdf]**
*submitted on 2018-06-08 12:57:27*

**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

[3] **viXra:1806.0108 [pdf]**
*submitted on 2018-06-08 15:43:03*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

JYB4, a logic system named after its writer Jean-Yves Béziau, is less a logic system and more of a model checking system based on 13 axioms for which p and q are assigned ±1 to evaluate models by arithmetic.
AR4, described by its writer Fabian Schang as a doxistic "deviant" logic system, is a follow-on of six more axioms.
Of the 19 axioms, 8 or 42% are not tautologous. We conclude that those statistics remove JYB4 and AR4 from further serious consideration as viable modern modal four-valued logics.

**Category:** Set Theory and Logic

[2] **viXra:1806.0036 [pdf]**
*submitted on 2018-06-04 15:31:06*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. info@cec-services dot com

"Possibly a cardinal number (index), within the set of all natural numbers, for the set of real numbers is greater than a cardinal number (index), within the set of all natural numbers, for the set of natural numbers."
"No cardinal number, within the set of all natural numbers, exists for a set greater than the cardinal number for the set of natural numbers and less than the cardinal number for the set of real numbers."
The mapping as rendered is not tautologous, and nearly a contradiction: FFFF FFFF FFFC FFFC.

**Category:** Set Theory and Logic

[1] **viXra:1806.0030 [pdf]**
*replaced on 2018-06-04 08:33:05*

**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