Relating First-Order Set Theories and Elementary Toposes

Steve Awodey,Carsten Butz,Alex Simpson,Thomas Streicher

doi:10.2178/bsl/1186666150

Abstract

AbstractWe show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo–Fraenkel set theory (IZF).

Highlights

The notion of elementary topos abstracts from the structure of the category of sets
The abstraction is sufficiently general that elementary toposes encompass a rich collection of other very different categories, including categories that have arisen in fields as diverse as algebraic geometry, algebraic topology, mathematical logic, and combinatorics
The reasoning supported in this way is both powerful and natural, it differs in several respects from the set-theoretic reasoning available in the familiar first-order set theories, such as Zermelo-Fraenkel set theory (ZF)

Summary

Introduction

The notion of elementary topos abstracts from the structure of the category of sets. The abstraction is sufficiently general that elementary toposes encompass a rich collection of other very different categories, including categories that have arisen in fields as diverse as algebraic geometry, algebraic topology, mathematical logic, and combinatorics. This additional structure, a directed structural system of inclusions (dssi), directly implements a well-behaved notion of subset relation between objects of a topos, not natural from a category-theoretic point of view, the structure of a dssi turns out to be exactly what is needed to obtain an interpretation of the full language of first-order set theory in a topos, including unbounded quantification; and resolves issue 2 above. We obtain a completeness result (Theorem 4.2) which shows that the theory BIST + Coll axiomatizes exactly the set-theoretic properties validated by our forcing semantics.

Results

Conclusion