Disproof of the Continuum Hypothesis and Determination of the Cardinality of Continuum by Approximations of Sets

TL;DR

This paper develops a set theory based on approximations, disproves the Continuum Hypothesis, and characterizes the cardinalities of sets without relying on the Axiom of Choice.

Contribution

It introduces a new set theory (AS) that invalidates some ZF axioms, disproves the Continuum Hypothesis, and provides a novel hierarchy of set cardinalities.

Findings

The Continuum Hypothesis is false in AS.

Sets with perfect subsets have maximal cardinality.

Cardinalities are always comparable without the Axiom of Choice.

Abstract

A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid. All the other axioms of ZF are valid and all the basic sets, such as complement, intersection and cartesian product, exist although complement is not quite the same set as in ZF. The set of all sets can be equipped with the topology of approximations (Ta). Every set is closed and every function is continuous in Ta. This implies that the Continuum Hypothesis is false. The sets containing a subset which is perfect in Ta are of the greatest cardinality. A simple observation shows that the concept of well-ordering must be defined in a slightly different way than in ZF. We prove that a set can be well-ordered if and only if it has no perfect subset. Therefore the cardinalities of arbitrary sets are always comparable without assuming the Axiom of Choice. The cardinals following the smallest infinite cardinal w are 2w, 3w, ..., w^2, 2w^2, 3w^2, >..., w^3.... each being of greater cardinality than the previous one, which is not the case in ZF. Immediately after these cardinals does not follow w^w which is not a well-orderable set but some well-ordered cardinal k, and this one is followed by the cardinals 2k, 3k, ..., k^2, 2k^2, 3k^2, ..., k^3...., etc. The greatest cardinal is P(w) and is not a well-orderable set. The cofinality of a well-ordered set is either 2 or w. The only regular cardinals are 0, 1, 2, w and P(w). All other cardinals are singular. The only strong limit cardinal is w. The only inaccessible cardinal is P(w). Strongly inaccessible cardinals do not exist.