We introduce a predicative version of topos (stratified pseudotopos) based on the notion of
small maps in algebraic set theory, developed by Joyal and one of the authors. Examples of …

We discuss a notion of universe in toposes which from a logical point of view gives rise to an
extension of Higher Order Intuitionistic Arithmetic (HAH) that allows one to construct families …

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 …

We propose a notion of interval object in a category with finite products, providing a
universal property for closed and bounded real line segments. The universal property gives …

We extend the usual internal logic of a (pre) topos to a more general interpretation, called
the stack semantics, which allows for" unbounded" quantifiers ranging over the class of …

According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm
independently of the type instantiations at which it is applied. The notion of relational …

This paper provides a unifying axiomatic account of the interpretation of recursive types that
incorporates both domain-theoretic and realizability models as concrete instances. Our …

This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the
necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I …

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 …

We show how to interpret the language of first-order set theory in an elementary topos
endowed with, as extra structure, a directed structural system of inclusions (dssi). As our …