Set Theory and Logic

1811 Submissions

[14] viXra:1811.0489 [pdf] submitted on 2018-11-28 09:23:10

"Refutation of Intuitionistic Logic on a "Transfinite Argument""

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate intuitionistic logic via Hilbert's "transfinite argument" and Komogorov's implementation. None of the axioms is tautologous.
Category: Set Theory and Logic

[13] viXra:1811.0442 [pdf] submitted on 2018-11-27 19:36:39

Refutation of the Two-Sided Page Paradox

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the two-sided page conjecture that: if either the front page implies the back page is false or the back page implies the front page is true is a paradox (contradiction). The conjecture is a theorem and hence refuted as a paradox.
Category: Set Theory and Logic

[12] viXra:1811.0439 [pdf] replaced on 2018-12-24 09:07:26

Remark on Seven Applications of Neutrosophic Logic: in Cultural Psychology, Economics Theorizing, Conflict Resolution, Philosophy of Science, Etc.

Authors: Victor Christianto, Florentin Smarandache
Comments: 12 Pages. This paper has been submitted to J. MATHEMATICS. Your comments are welcome

In this short communication, we review seven applications of NFL which we have explored in a number of papers: 1) Background: The purpose of this study is to review on how Neutrosophic Logic can be found useful in a number of diverse areas of interest; 2) Methods: we use logical analysis based on NL; 3) Results: Some fields of study may be found elevated after analyzed by NL theory; and 4) Conclusions: We can expect NL theory can be applied in many areas of research too, both in applied mathematics, economics, and also physics. Hopefully the readers will find a continuing line of thoughts in our research in the last few years
Category: Set Theory and Logic

[11] viXra:1811.0382 [pdf] submitted on 2018-11-23 05:59:45

Refutation of Two Modern Modal Logics: "JYB4" and the Follow-on "AR4"

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the four-valued logic systems of J.-Y. Béziau and F. Schang as JYB4 and AR4. JYB4 named the four-valued modal logic Ł4 as Łukasiewicz's nightmare because of the alleged absurdity of (◇p&◇q) → ◇(p&q). A model checking system is then framed based on 0± and 1±. We show (◇p&◇q) → ◇(p&q) is equivalent to (◇p&◇~p) → ◇(p&~p) with (◇p&◇q) = ◇(p&q) as a theorem. AR4 was a doxastic logic follow-on to JYB4. We name these modern modal logic systems as Béziau's nightmare and Schang's nightmare.
Category: Set Theory and Logic

[10] viXra:1811.0374 [pdf] submitted on 2018-11-23 13:43:15

Rule of Necessitation: a Non-Contingent Truthity, But not a Tautology

Authors: Colin James III
Comments: 4 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

1. The axiom or rule of necessitation N states that if p is a theorem, then necessarily p is a theorem: If ⊢ p then ⊢ ◻p. We show this is non-contingent (a truthity), but not tautologous (a proof). We evaluate axioms (in bold) of N, K, T, 4, B, D, 5 to derive systems (in italics) of K, M, T, S4, S5, D. We conclude that N the axiom or rule of necessitation is not tautologous. Because system M as derived and rendered is not tautologous, system G-M also not tautologous. What follows is that systems derived from using M are tainted, regardless of the tautological status of the result so masking the defect, such as systems S4, B, and S5. We also find that Gentzen-sequent proof is suspicious, perhaps due to its non bi-valent lattice basis in a vector space.
Category: Set Theory and Logic

[9] viXra:1811.0269 [pdf] submitted on 2018-11-17 13:14:25

Refutation of Superposition as Glue in Matita Theorem Prover

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the substitution lemma for the successor function, smart application of inductive hypotheses, and proof traces of a complex example in the Matita standard library. Results are not tautologous, hence refuting superposition.
Category: Set Theory and Logic

[8] viXra:1811.0264 [pdf] submitted on 2018-11-17 18:17:31

Refutation of Metamath Theorem Prover

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate six conjectures and one theorem, as proffered by Metamath staff. The conjectures are not tautologous. The Tarski-Grothendieck theorem is also not tautologous. Metamath fails our analysis.
Category: Set Theory and Logic

[7] viXra:1811.0220 [pdf] submitted on 2018-11-14 18:40:03

Shorter Refutation of the Löb Theorem and Gödel Incompleteness by Substitution of Contradiction

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Löb’s theorem □(□X→X)→□X and Gödel’s incompleteness as □(□⊥ →⊥)→□⊥ are refuted.
Category: Set Theory and Logic

[6] viXra:1811.0154 [pdf] submitted on 2018-11-09 08:30:47

Find the extra shape (In Russian)

Authors: V. A. Kasimov
Comments: 2 Pages. in Russian

First, the task will put you in a dead end. No figure clearly stands out from the total number more than others. It's not as easy as it may seem in the first seconds.
Category: Set Theory and Logic

[5] viXra:1811.0078 [pdf] submitted on 2018-11-05 15:44:40

Refutation of Behavioral Mereology

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

If P≤P′ and Q′≤Q, proposition <>P'P<>QP' = <>QP is equivalent to []P'P[]QP' = []QP and respectively not tautologies.
Category: Set Theory and Logic

[4] viXra:1811.0075 [pdf] submitted on 2018-11-05 20:42:06

Refutation of the Blok-Esakia Theorem for Universal Classes

Authors: Colin James III
Comments: 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Grzegorczyk (gzr) algebras as used for support and the Blok-Esakia theorems are not confirmed as tautologies and hence refuted.
Category: Set Theory and Logic

[3] viXra:1811.0059 [pdf] submitted on 2018-11-04 19:11:29

Refutation of First-Order Continuous Induction on Real Closed Fields

Authors: Colin James III
Comments: 3 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

By mapping definitions, theorems, and propositions into Meth8/VŁ4, we refute the first-order continuous induction principle on real closed fields.
Category: Set Theory and Logic

[2] viXra:1811.0020 [pdf] submitted on 2018-11-01 10:35:55

New Axioms in the Set Theory

Authors: Antoine Balan
Comments: 2 pages, written in english

We propose three new axioms in set theory, axiomatising the measure theory of the Hilbert spaces.
Category: Set Theory and Logic

[1] viXra:1811.0018 [pdf] replaced on 2019-01-21 17:31:05

Elementary Set Theory Used To Prove FLT

Authors: Phil A. Bloom
Comments: 2 Pages.

An open problem is proving FLT \emph{simply} (using Fermat's toolbox) for each $n\in\mathbb{N}, n>2$. Our \emph{direct proof} (not BWOC) of FLT is based on our algebraic identity $((r+2q^n)^\frac{1}{n})^n-((r-2q^n)^\frac{1}{n})^n=(2^\frac{2}{n}q)^n$ with arbitrary values of $n\in\mathbb{N}$, and with $r\in\mathbb{R},q\in\mathbb{Q},n,q,r>0$. For convenience, we \emph{denote} $(r+2q^n)^\frac{1}{n}$ by $s$; we \emph{denote} $(r-2q^n)^\frac{1}{n}$ by $t$; and, we \emph{denote} $2^\frac{2}{n}q$ by $u$. For any given $n>2$ : Since the term $u$ or $2^\frac{2}{n}q$ with $q\in\mathbb{Q}$ is not rational, this identity allows us to relate null sets $\{(s,t,u)|s,t,u\in\mathbb{N},s,t,u>0,s^n-t^n=u^n\}$ with subsequently proven null sets $\{z,y,x|z,y,x\in\mathbb{N},z,y,x>0,z^n-y^n=x^n\}$. We show it is true, for $n>0$, that $\{u|s,t,u\in\mathbb{N},s,t,u>0,s^n-t^n=u^n\}=\{x|z,y,x\in\mathbb{N},z,y,x>0,z^n-y^n=x^n\}$. Hence, for any given $n\in\mathbb{N},n>2$, it is a true statement that $\{(x,y,z)|x,y,z\in\mathbb{N},x,y,z>0,x^n+y^n=z^n\}=\varnothing$.
Category: Set Theory and Logic