Algebraic Set Theory (London Mathematical Society Lecture Note Series, Series Number 220)

This book offers a new, algebraic, approach to set theory. The authors introduce a particular kind of algebra, the Zermelo-Fraenkel algebras, which arise from the familiar axioms of Zermelo-Fraenkel set theory. Furthermore the authors explicitly construct such algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory. In particular it provides a uniform description of various constructions of the cumulative hierarchy of sets in forcing models, sheaf models and realisability models. Graduate students and researchers in mathematical logic, category theory and computer science should find this book of great interest, and it should be accessible to anyone with some background in categorical logic.

This book offers a new algebraic approach to set theory. The authors introduce a particular kind of algebra, the Zermelo-Fraenkel algebras, which arise from the familiar axioms of Zermelo-Fraenkel set theory. Furthermore, the authors explicitly construct these algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory. In particular it provides a uniform description of various constructions of the cumulative hierarchy of sets in forcing models, sheaf models and realizability models. Graduate students and researchers in mathematical logic, category theory and computer science should find this book of great interest, and it should be accessible to anyone with a background in categorical logic.

Cover; Title; Copyright; Contents; Preface; Introduction; I Axiomatic Theory of Small Maps; 1 Axioms for small maps; 2 Representable structures; 3 Power-sets; 4 Complete sup-lattices; 5 Appendix: Uniqueness of universal small maps; II Zermelo-Fraenkel Algebras; 1 Free Zermelo-Fraenkel algebras; 2 Ordinal numbers; 3 Von Neumann ordinals; 4 The Tarski fixed point theorem; 5 Axioms for set theory; III Existence Theorems; 1 Open maps and (bi- )simulations; 2 Forests; 3 Height functions; 4 Construction of V and 0; 5 Construction of Tarski ordinals 6 Simulation for Von Neumann ordinalsIV Examples; 1 Sets and classes; 2 Kuratowski finite maps; 3 Sheaves on a site; 4 Readability; 5 Choice maps; Appendix A. Monads and algebras with successor; Appendix B. Heyting pretopoi; Appendix C. Descent; Bibliography; Index