Algebraic Set Theory

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Cambridge University Press, Sep 14, 1995 - Mathematics - 123 pages
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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.
  

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Contents

Introduction
1
Axiomatic Theory of Small Maps
7
2 Representable structures
11
3 Powersets
16
4 Complete suplattices
22
Uniqueness of universal small maps
24
ZermeloFraenkel Algebras
29
2 Ordinal numbers
38
4 Construction of V and O
77
5 Construction of Tarski ordinals
81
6 Simulation for Von Neumann ordinals
83
Examples
87
2 Kuratowski finite maps
88
3 Sheaves on a site
89
4 Realizability
92
5 Choice maps
96

3 Von Neumann ordinals
46
4 The Tarski fixed point theorem
54
5 Axioms for set theory
59
Existence Theorems
67
2 Forests
71
3 Height functions
74
Monads and algebras with successor
101
Hey ting pretopoi
109
Descent
113
Bibliography
117
Index
120
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