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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\u2013Fraenkel set theory (IZF).<\/jats:p>","DOI":"10.2178\/bsl\/1186666150","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T14:56:48Z","timestamp":1197557808000},"page":"340-358","source":"Crossref","is-referenced-by-count":10,"title":["Relating First-Order Set Theories and Elementary Toposes"],"prefix":"10.1017","volume":"13","author":[{"given":"Steve","family":"Awodey","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carsten","family":"Butz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Simpson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thomas","family":"Streicher","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600002067_ref011","volume-title":"Introduction to higher order categorical logic","author":"Lambek","year":"1986"},{"key":"S1079898600002067_ref009","volume-title":"Sketches of an elephant: a topos theory compendium","volume":"1\u20132","author":"Johnstone","year":"2002"},{"key":"S1079898600002067_ref017","first-page":"77","volume-title":"Proceedings of the 14th annual IEEE symposium on logic in computer science","author":"Simpson","year":"1999"},{"key":"S1079898600002067_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090976"},{"key":"S1079898600002067_ref018","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2003.12.005"},{"key":"S1079898600002067_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0927-0"},{"key":"S1079898600002067_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(02)00259-1"},{"key":"S1079898600002067_ref020","volume-title":"Constructivism in mathematics","volume":"I\u2013II","author":"Troelstra","year":"1988"},{"key":"S1079898600002067_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(00)00031-2"},{"key":"S1079898600002067_ref004","doi-asserted-by":"crossref","unstructured":"Awodey S. , Butz C. , Simpson A.K. , and Streicher T. , Relating first-order set theories, toposes and categories of classes, in preparation, 2007.","DOI":"10.2178\/bsl\/1186666150"},{"key":"S1079898600002067_ref015","doi-asserted-by":"publisher","DOI":"10.2307\/2272159"},{"key":"S1079898600002067_ref002","volume-title":"Notes on constructive set theory","author":"Aczel","year":"2001"},{"key":"S1079898600002067_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(80)90096-1"},{"key":"S1079898600002067_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71989-X"},{"key":"S1079898600002067_ref019","volume-title":"Constructive set theories and their category-theoretic models. 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