[10] **viXra:1712.0674 [pdf]**
*replaced on 2018-01-01 16:39:21*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2017 by Colin James III All rights reserved.

A recent advance is that Meth8/VŁ4 finds 0\1 to be undefined, instead of 0.
What follows is that zero is not a natural number as commonly used.
We generalize Eqs. 1.1-1.4 from 0 and 1 onto 0 and n.
eneralized Eqs. 2.1-2.4 match the results of Eqs. 1.1-1.4, confirming consistency.

**Category:** Set Theory and Logic

[9] **viXra:1712.0600 [pdf]**
*submitted on 2017-12-26 09:29:23*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2017 by Colin James III All rights reserved

Eqs. 1.2 and 1.4 as rendered are not tautologous. Hence Meth8/VL4 finds LEMI suspicious.

**Category:** Set Theory and Logic

[8] **viXra:1712.0599 [pdf]**
*submitted on 2017-12-26 09:31:05*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2017 by Colin James III All rights reserved.

Eq. 2.2 as rendered is a contradiction, hence Eq. 2.1 as INF is suspicious.

**Category:** Set Theory and Logic

[7] **viXra:1712.0595 [pdf]**
*submitted on 2017-12-26 00:23:47*

**Authors:** Temur Z. Kalanov

**Comments:** 17 Pages.

The critical analysis of the foundation of set theory is proposed. The unity of formal logic and rational dialectics is the correct methodological basis of the analysis. The analysis leads to the following results: (1) the mathematical concept of set should be analyzed on the basis of the formal-logical clauses “Definition of concept”, “Logical class”, “Division of concept”, “Basis of division”, “Rules of division”; (2) the standard mathematical theory of sets is an erroneous theory because it does not contain definition of the concept “element (object) of set”; (3) the concept of empty set (class) is a meaningless, erroneous, and inadmissible one because the definition of the concept “empty set (class)” contradicts to the definition of the logical class. (If the set (class) does not contain a single element (object), then there is no feature (sign) of the element (object). This implies that the concept of empty set (class) has no content and volume (scope). Therefore, this concept is inadmissible one); (4) the standard mathematical operations of union, intersection and difference of sets (classes) are meaningless, erroneous and inadmissible operations because they do not satisfy the following formal-logical condition: every separate element (object) of the set (class) must be in only one some set (class) and cannot be in two sets (classes). Thus, the results of formal-logical analysis prove that the standard mathematical theory of sets is an erroneous theory because it does not satisfy the criterion of truth.

**Category:** Set Theory and Logic

[6] **viXra:1712.0403 [pdf]**
*replaced on 2017-12-23 02:39:19*

**Authors:** Jaykov Foukzon

**Comments:** 14 Pages. Journal of Global Research in Mathematical Archives (JGRMA)Vol 5, No 1 (2018): January-2018

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st),(ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k),[10],11].

**Category:** Set Theory and Logic

[5] **viXra:1712.0386 [pdf]**
*submitted on 2017-12-12 01:26:41*

**Authors:** Liguo Fei, Yong Deng

**Comments:** 21 Pages.

Pythagorean fuzzy set (PFS) initially extended by Yager from intuitionistic fuzzy set (IFS), which can model uncertain information with more general conditions in the process of multi-criteria decision making (MCDM). The fuzzy decision analysis of this paper is mainly based on
two expressions in Pythagorean fuzzy environment, namely, Pythagorean fuzzy number (PFN) and interval-valued Pythagorean fuzzy number (IVPFN). We initiate a novel axiomatic definition of Pythagorean fuzzy distance measure, including PFNs and IVPFNs, and put forward the corresponding theorems and prove them. Based on the defined distance measures, the closeness indexes are developed for PFNs and IVPFNs inspired by the idea of technique for order preference by similarity to ideal solution (TOPSIS) approach. After these basic definitions have been established, the
hierarchical decision approach is presented to handle MCDM problems under Pythagorean fuzzy environment. To address hierarchical decision issues, the closeness index-based score function is defined to calculate the score of each permutation for the optimal alternative. To determine criterion weights, a new method based on the proposed similarity measure and aggregation operator of PFNs and IVPFNs is presented according to Pythagorean fuzzy information from decision
matrix, rather than being provided in advance by decision makers, which can effectively reduce human subjectivity. An experimental case is conducted to demonstrate the applicability and flexibility of the proposed decision approach. Finally, the extension forms of Pythagorean fuzzy decision approach for heterogeneous information are briefly introduced as the further application in other uncertain information processing fields.

**Category:** Set Theory and Logic

[4] **viXra:1712.0368 [pdf]**
*submitted on 2017-12-09 22:30:01*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2017 by Colin James III All rights reserved.

"[O]one cannot infer from X simulates Y, and Y has property P, to the conclusion that therefore X has Y's property P for arbitrary P." is tautologous.
"The contrapositive of the inference is logically equivalent—X simulates Y, X does not have P therefore Y does not [have P]" is not tautologous.
The two conjectuses are not logically equivalent.
Hence the brain Simulator reply of the Chinese room argument is confirmed and validated.

**Category:** Set Theory and Logic

[3] **viXra:1712.0205 [pdf]**
*submitted on 2017-12-06 16:42:00*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We map the argument for Cantor's diagonal argument into the Meth8 modal logic model checker.
The two main equations as rendered are not tautologous. Hence Cantor's diagonal argument is not supported.

**Category:** Set Theory and Logic

[2] **viXra:1712.0204 [pdf]**
*submitted on 2017-12-06 17:11:12*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2016 by Colin James III All rights reserved.

Refutation of the Gödel-Löb Axiom implies the refutation of the Axiom of Choice. Both axioms as separately rendered are also not tautologous.

**Category:** Set Theory and Logic

[1] **viXra:1712.0139 [pdf]**
*replaced on 2018-04-03 05:31:20*

**Authors:** Johan Noldus

**Comments:** 1 Page.

We show that the axiom of choice is false.

**Category:** Set Theory and Logic