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A brief introduction to algebraic set theory Steve Awodey Abstract This brief article is intended to introduce the reader to the field of algebraic set theory in which models of set theory of a new and fascinating kind are determined algebraically The method is quite robust applying to various classical intuitionistic and constructive set theories Under this scheme some familiar set theoretic properties are related to algebraic ones while others result from logical constraints Conventional elementary set theories are complete with respect to algebraic models which arise in a variety of ways including topologically type theoretically and through variation Many previous results from topos theory involving realizability permutation and sheaf models of set theory are subsumed and the prospects for further such unification seem bright 1 Introduction Algebraic set theory AST is a new approach to the construction of models of set theory invented by Andre Joyal and Ieke Moerdijk and first presented in detail in 31 It promises to be a flexible and powerful tool for the investigation of classical and intuitionistic systems of elementary set theory bringing to bear a new insight into the models of such systems Indeed it has already proven to be a quite robust framework applying to the study of classical intuitionistic bounded and predicative systems and subsuming some previously unrelated techniques The new insight taken as a starting point in AST is that models of set theory are in fact algebras for a suitably presented algebraic theory and that many familiar set theoretic conditions such as well foundedness are thereby related to familiar algebraic ones such as freeness AST is currently the focus of active research by several authors and new methods are being developed for the construction and organization of models of various different systems as well as for relating this approach with other more traditional ones Some recent results are mentioned here however the aim is not

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