Deciding the Continuum Hypothesis with the Inverse Powerset

TL;DR

This paper introduces inverse powersets within an extended set theory framework, providing new axioms that relate to the continuum hypothesis and exploring the concept of empty sets with varying cardinalities.

Contribution

It proposes a novel extension to Zermelo-Fraenkel set theory by defining inverse powersets and investigates their implications for the continuum hypothesis.

Findings

One extension implies the continuum hypothesis.

Another extension implies its negation.

Explores empty sets with different cardinalities.

Abstract

We introduce the concept of inverse powerset by adding three axioms to the Zermelo-Fraenkel set theory. This extends the Zermelo-Fraenkel set theory with a new type of set which is motivated by an intuitive meaning and interesting applications. We present different ways to extend the definition of cardinality and show that one implies the continuum hypothesis while another implies the negation of the continuum hypothesis. We will also explore the idea of empty sets of different cardinalities which could be seen as the empty counterpart of Cantor's theorem for infinite sets.