Abstract. 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:

The axiomofchoice and the law of excluded middle in weak set theories

Abstract not found

Abstract. We extract a quantitative variant of uniqueness from the usual hypotheses of the implicit functions theorem. This leads not only to an a priori proof of continuity, but also to an alternative, fully constructive existence proof. 1

This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals ” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are