## Relating first-order set theories, toposes and categories of classes (2006)

Venue: | In preparation |

Citations: | 3 - 3 self |

### BibTeX

@INPROCEEDINGS{Awodey06relatingfirst-order,

author = {Steve Awodey and Thomas Streicher},

title = {Relating first-order set theories, toposes and categories of classes},

booktitle = {In preparation},

year = {2006}

}

### OpenURL

### 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. Since every topos is equivalent to one carrying a dssi, the language of first-order has a forcing interpretation in every elementary topos. 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 every cocomplete topos and every realizability topos can be endowed (up to equivalence) with such a superdirected structural system of inclusions. This provides a uniform explanation for why such “real-world ” toposes model Separation. 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 Moerdijk’s Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with ∗Corresponding author. 1Previously, lecturer at Heriot-Watt University (2000–2001), and the IT University of