A topological set theory implied by ZF and GPK

TL;DR

This paper introduces a new topological set theory that generalizes ZF, GPK, and hyperuniverses, maintaining their expressiveness and consistency strength while exploring the impact of additional axioms like the universal set.

Contribution

It proposes a unified axiomatic system motivated by topology that weakens but retains the core features of ZF, GPK, and hyperuniverses, and analyzes the effects of specific axioms.

Findings

The system has the same consistency strength as ZF.

Adding the universal set axiom increases strength to GPK.

Results are independent of the existence of the empty class and atoms.

Abstract

We present a system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK and the theory of hyperuniverses. On the other hand, it retains most of the expressiveness of these theories and has the same consistency strength as ZF. We single out the additional axiom of the universal set as the one that increases the consistency strength to that of GPK and explore several other axioms and interrelations between those theories. Our results are independent of whether the empty class is a set and whether atoms exist. This article is a revised version of the first part of the author's doctoral thesis.