Abstract
AbstractThis 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, such as 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.
Highlights
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]
The new insight taken as a starting point in AST is that models of set theory are algebras for a suitably presented algebraic theory, and that many familiar set theoretic conditions are thereby related to familiar algebraic ones
The list of references includes some works not cited in the text and should serve as a guide to the literature, which the reader will hopefully find more accessible in virtue of this brief introduction
Summary
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]. Like the original presentation by Joyal & Moerdijk, much of the research in AST involves a fairly heavy use of category theory Whether this is really essential to the algebraic approach to set theory could be debated; but just as in other “algebraic” fields like algebraic geometry, topology, and number theory, the convenience of functorial methods is irresistible and has strongly influenced the development of the subject
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