Set Theory and Logic

1802 Submissions

[4] viXra:1802.0306 [pdf] submitted on 2018-02-21 22:19:21

Refutation of Higher-Order Logic as Bivalent © 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.

We evaluate higher-order logic based on the principle of mathematical induction. Meth8/VŁ4 treats sets and variables as variables. The quantification over quantification is not bivalent. We alleviate this constraining condition by distributing the quantified expression over nested expressions. At each nested level, the quantification is explicitly distributed for clarity. We conclude that higher-order logic is not bivalent and that nested quantification is better expressed as explicitly distributed.
Category: Set Theory and Logic

[3] viXra:1802.0234 [pdf] submitted on 2018-02-18 18:17:32

Refutation of the Bertrand Postulate and Bertrand-Chebyshev 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.

[F]for every n > 1, there is always at least one prime p such that n < p < 2 n.(1.1) For all n∈N>0, there exists a prime number p with n<p≤2n.(2.1) Eqs. 1.2 and 2.2 as rendered are not tautologous, meaning both Bertrand expressions are suspicious.
Category: Set Theory and Logic

[2] viXra:1802.0182 [pdf] submitted on 2018-02-15 08:26:17

Prenex Normal Form with Prefix and Matrix Refuted as not Bivalent © 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.

We evaluate prenex normal form of quantifier presentation on rules for the connectives of conjunction, disjunction, implication, and for negation. The format is not tautologous, not bivalent, and hence refuted. What follows is that many theorems produced with prenex for computer science, mathematics, and physics are now suspicious. A notable example is the satisfiability algorithms produced by Martin Davis and Hilary Putnam which are now mistaken.
Category: Set Theory and Logic

[1] 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