Set Theory and Logic

2304 Submissions

[2] viXra:2304.0140 [pdf] submitted on 2023-04-19 01:07:16

Contradiction and Reconstruction of Axiom of ZF System

Authors: Jincheng Zhang
Comments: 19 Pages.

In set theory ZF system, there are specious strange conclusions when set of infinite sets are interpreted into numbers. Axiom of infinity in ZF system confuses the difference between δ+1＝δ and δ+1≠δ, Error of Cantor's set theory is the error of axiom of infinity. We use (δ + 1 = δ or δ + 1 ≠ δ) to establish two opposite sets of axioms SZF + and SZF-, which are contradictory systems, similar to Euclidean geometry and non-Euclidean geometry axioms.In set theory SZF system, after correcting the axiom of infinity, we can prove that "for any definable predicate P (n), if it is true to finite n, then for any infinite number a, P(a) is also true." Power set axiom, axiom of separation, axiom of substitution, axiom of choice, axiom of foundation these axioms have become provable propositions.In fact, the four axioms and several definitions of "axiom of empty set, set axioms, union axioms, and infinite axioms" are sufficient for the set theory system.In Cantor’s set theory, there are ordinal numbers and uncountable ordinal numbers. There are no uncountable ordinal numbers, so proof of uncountable ordinal number is false. The transfinite induction in Cantor’s set theory is also false; in fact, transfinite induction is very simple, we can prove that: for any definable predicate P (n), if it is true to finite n, then for any infinite number a, P(a) is also true. In classical set theory system, all difficulties are from infinite difficulties. After we have solved the axiom of infinity, infinity is as simple as finite.
Category: Set Theory and Logic

[1] viXra:2304.0008 [pdf] replaced on 2023-06-03 23:37:58

ZFC Inconsistency Explained and Defended

Authors: Jim Rock
Comments: 1 Page.

There is a class of sets that can be constructed within ZFC that both have and do not have a largest element. Two contradictory arguments about these properties are developed and defended.
Category: Set Theory and Logic