An axiomatic approach to higher order set theory

TL;DR

This paper introduces an axiomatic framework for countable higher order set theory, establishing its consistency relative to ZFC with inaccessible cardinals and demonstrating its applicability to areas like category theory.

Contribution

It presents a new axiomatic theory of collections for countable higher order set theory, linking it to established set theories and foundational for certain mathematical domains.

Findings

Equiconsistency with ZFC plus countable inaccessible cardinals

Provides a canonical foundation for parts of mathematics like category theory

Formalizes a 'theory of collections' for higher order set theory

Abstract

Higher order set theory has been a topic of interest for some time, with recent efforts focused on the strength of second order set theories [KW16]. In this paper we strive to present one 'theory of collections' that allows for a formal consideration of 'countable higher order set theory'. We will see that this theory is equiconsistent with $ZFC$ plus the existence of a countable collection of inaccessible cardinals. We will also see that this theory serves as a canonical foundation for some parts of mathematics not covered by standard set/class theories (e.g. $ZFC$ or $MK$), such as category theory.