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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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 2019-01-15 22:07:17*

**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 $((r+2q^n)^\frac{1}{n})^n-((r-2q^n)^\frac{1}{n})^n=(2^\frac{2}{n}q)^n$ for which $n$ is any given positive natural number, $r$ is positive real and $q$ is positive rational such that the set of triples $\{((r+2q^n)^\frac{1}{n},(r-2q^n)^\frac{1}{n},2^\frac{2}{n}q)\}$ is not empty with $(r+2q^n)^\frac{1}{n},(r-2q^n)^\frac{1}{n},(2^\frac{2}{n}q)\in\mathbb{N}$. We relate this set of triples to $\{z,y,x|z,y,x\in\mathbb{N}\}$ for which the transposed \emph{Fermat equation} $z^n-y^n=x^n$ holds. We demonstrate, for any given value of $n$, that $2^\frac{2}{n}q=x$. Clearly, for $n>2$, the term $2^\frac{2}{n}q$ with $q\in\mathbb{Q}$ is not rational. Consequently, for values of $n\in\mathbb{N},n>2$, it is true that $\{(x,y,z)|x,y,z\in\mathbb{N},x^n+y^n=z^n\}=\varnothing$.

**Category:** Set Theory and Logic