This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on Realisability  and the other on Sheaf Models in Algebraic Set Theory . Sheaf theory and realisability have been effective methods for constructing models of various constructive and intuitionistic… (More)
1 Introduction This paper is the second in a series on the relation between algebraic set theory  and predicative formal systems. The purpose of the present paper is to show how realizability models of constructive set theories fit into the framework of algebraic set theory. It can be read independently from the first part ; however, we recommend… (More)
We show how one may establish proof-theoretic results for constructive Zermelo-Fraenkel set theory, such as the compactness rule for Cantor space and the Bar Induction rule for Baire space, by constructing sheaf models and using their preservation properties.