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 main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full (...) axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo—Fraenkel set theory (IZF). (shrink)

_Stefania Centrone, Deborah Kant_, and _Deniz Sarikaya_, eds, _ Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory, and General Thoughts _. Studies in Epistemology, Logic, Methodology, and Philosophy of Science; 407. Springer, 2019. Pp. xxviii + 494. ISBN: 978-3-030-15654-1 ; 978-3-030-15655-8. doi.org/10.1007/978-3-030-15655-8† †.