Inaccessible set axioms may have little consistency strength

Abstract

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form V κ where κ is a strongly inaccessible cardinal and V κ denotes the κth level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depend on the context in which they are embedded. The context here will be the theory CZF − of constructive Zermelo–Fraenkel set theory but without ∈-Induction (foundation). Let INAC be the statement that for every set there is an inaccessible set containing it. CZF −+ INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF −+ INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF −+ INAC is a small ordinal known as the Feferman–Schütte ordinal Γ 0 .