 
Summary: Aspects of Predicative Algebraic Set Theory II:
Realizability
Benno van den Berg & Ieke Moerdijk
January 15, 2008
Dedicated to JeanYves Girard on the occasion of his 60th birthday
1 Introduction
This paper is the second in a series on the relation between algebraic set theory
[19] 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 [5];
however, we recommend that readers of this paper read the introduction to [5],
where the general methods and goals of algebraic set theory are explained in
more detail.
To motivate our methods, let us recall the construction of Hyland's effective
topos Eff [17]. The objects of this category are pairs (X, =), where = is a subset
of N × X × X satisfying certain conditions. If we write n x = y in case the
triple (n, x, y) belongs to this subset, then these conditions can be formulated
by requiring the existence of natural numbers s and t such that
s x = x x = x
t x = x x = x x = x .
