Constructive toposes with countable sums as models of constructive set theory

Heyting-valued interpretations for Constructive Set Theory

The theory of locales [23] has a twofold interplay with intuitionistic mathematics: first of all, the internal logic of toposes and intuitionistic set theories provide suitable settings for the development of the theory of locales [24], and secondly, the notion of a locale determines two important forms of toposes and of interpretations for intuitionistic set theories, namely localic toposes [2...

Type theories, toposes and constructive set theory: predicative aspects of AST

The Relation Reflection Scheme

In this paper we introduce a new axiom scheme, the Relation Reflection Scheme (RRS), for constructive set theory. Constructive set theory is an extensional set theoretical setting for constructive mathematics. A formal system for constructive set theory was first introduced by Myhill in [8]. In [1, 2, 3] I introduced a formal system CZF that is closely related to Myhill’s formal system and gave...

Variations on Realizability: Realizing the Propositional Axiom of Choice

Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations to establish consistency, independence and conservativity results. van Oosten’s contribution to the Workshop (see van Oos...

Arithmetic universes and classifying toposes

The paper uses structures in Con, the author’s 2-category of sketches for arithmetic universes (AUs), to provide constructive, base-independent results for Grothendieck toposes (bounded S-toposes) as generalized spaces. The main result is to show how an extension map U : T1 → T0 can be viewed as a bundle, transforming base points (models of T0 in any elementary topos S with nno) to fibres (gene...