Predicative Algebraic Set Theory

In this paper the machinery and results developed in [Awodey et al, 2004] are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certain structure. Specifically, basic categories of classes and categories of classes are axiomatized and shown to provide models of the aforementioned set theories. Finally, models of these theories are constructed in the category of ideals. The purpose of this paper is to generalize the machinery and results developed by Awodey, Butz, Simpson and Streicher in [Awodey et al, 2004] to the predicative case. Specifically, in ibid. it was shown that: 1. every category of classes contains a model of the intuitionistic, elementary set theory BIST, 2. BIST is logically complete with respect to such class category models, 3. the category of sets in such a model is an elementary topos, 4. every topos occurs as the sets in such a category of classes. It follows, in particular, that BIST is sound and complete with respect to topoi as they can occur in categories of classes. Thus, in a very precise sense, BIST represents exactly the elementary set theory whose models are the elementary topoi. In the current paper, we show that the same situation obtains with respect to a weaker, predicative, set theory CST which lacks the powerset axiom, and the new notion of a predicative topos (called a Π-pretopos, and defined as a locally cartesian closed pretopos). As in the impredicative case, the correspondence between the set theory and the category is mediated by a suitable category of classes, now weakened by the omission of the small We wish to acknowledge many helpful discussions with Carsten Butz, Henrik Forssell, Nicola Gambino, André Joyal, Ivar Rummelhoff, Dana Scott, Thomas Streicher, and especially Alex Simpson. Received by the editors 2004-12-27. Published on 2005-04-12 in the volume of articles from CT2004. 2000 Mathematics Subject Classification: 18B05, 18B25, 18C10, 03G30, 03E70, 03F60.