Abstract
This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to one carrying a dssi. We prove that the set theory BIST+Coll (where Coll is the strong Collection axiom) is sound and complete relative to forcing interpretations in toposes with natural numbers object (nno). Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can (up to equivalence) be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijkʼs Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation.
Highlights
The notion of elementary topos abstracts from the structure of the category of sets, retaining many of its essential features
The reasoning supported by the internal logic is both natural and powerful, but 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)
In Part I of the paper, we present the set theory that we shall interpret over an aribitrary elementary topos, which we call Basic Intuitionistic Set Theory (BIST)
Summary
The notion of elementary topos abstracts from the structure of the category of sets, retaining many of its essential features. The forcing interpretation and construction of the category of ideals can be defined for any topos, as claimed above The proof of this uses a notion of membership graph, which adapts the transitive objects developed by Cole, Mitchell and Osius, see [34, 14, 40, 30], to a set-theoretic universe incorporating (a class of) atoms. The paper concludes with a short section which discusses the relation of our work to other more recent research
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