Comparing material and structural set theories

TL;DR

This paper compares structural set theories based on category theory with traditional material set theories, analyzing their axioms and formal relationships.

Contribution

It introduces category-theoretic versions of strong set-theoretic axioms and compares them to classical membership-based theories.

Findings

Category-theoretic axioms for set theories are formally comparable to traditional ones.

Weak intuitionistic and predicative theories of pretoposes are developed.

A classical construction relates internal well-founded relations to material set theories.

Abstract

We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and predicative theories of pretoposes, and we also propose category-theoretic versions of stronger axioms such as unbounded separation, replacement, and collection. Finally, we compare all of these theories formally to traditional membership-based or "material" set theories, using a version of the classical construction based on internal well-founded relations.