Set Theory and Logic

1712 Submissions

[6] viXra:1712.0403 [pdf] replaced on 2017-12-15 05:52:44

There is no Standard Model of ZFC

Authors: Jaykov Foukzon
Comments: 13 Pages.

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

On Multi-Criteria Pythagorean Fuzzy Decision-Making

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

The Brain Simulator Reply (BSR) of the Chinese Room Argument (Cra) is Confirmed. © Copyright 2017 by Colin James III All Rights Reserved.

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

Refutation of Cantor's Diagonal Argument © Copyright 2017 by Colin James III All Rights Reserved.

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

Refutation of Axiom of Choice Via Refutation of the Gödel-Löb Axiom © Copyright 2016 by Colin James III All Rights Reserved.

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] submitted on 2017-12-06 12:44:00

A Proof of the Falsity of the Axiom of Choice.

Authors: Johan Noldus
Comments: 1 Page.

We show that the axiom of choice is false.
Category: Set Theory and Logic