[10] **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

[9] **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

[8] **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

[7] **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

[6] **viXra:1811.0148 [pdf]**
*replaced on 2018-11-14 07:22:11*

**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.

In Prover9 the alleged Łukasiewicz nightmare of (◇p&◇q)→◇(p&q) is not tautologous. However, in Prover 9 the nightmare recast in one variable as (◇p&◇~p)→◇(p&~p) is tautologous. In Meth8/VŁ4, both propositions are tautologous. This speaks to Meth8/VŁ4, based on the corrected modern Square of Opposition as an exact bivalent system, opposed to Prover9, based on the uncorrected modern Square of Opposition as an inexact probabilistic vector space.

**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 2018-11-09 17:10:10*

**Authors:** Phil A. Bloom

**Comments:** 3 Pages.

An open problem is proving FLT simply for each integral $n>2$. Our proof of FLT is based on our algebraic identity, denoted, {for convenience}, as $r^n+s^n=t^n$. For $n\geq1$ we relate $r,s,t>0$, each a different function of variables comprising $r^n+s^n=t^n$, with $x,y,z>0$ for which $x^n+y^n=z^n$ holds. We infer as true by \emph{direct argument} (not BWOC), for any given $n>2$, that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}$. In addition, we show, for $n>2$, that $\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}=\varnothing$. Thus, for $n\in\mathbb{Z},n>2$, it is true that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\varnothing$.

**Category:** Set Theory and Logic