Algebraic Set Theory
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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Axiomatic Theory of Small Maps
2 Representable structures
4 Complete suplattices
Uniqueness of universal small maps
2 Ordinal numbers
4 Construction of V and O
5 Construction of Tarski ordinals
6 Simulation for Von Neumann ordinals
2 Kuratowski finite maps
3 Sheaves on a site
5 Choice maps
ambient category arrow axiom S3 axioms for small bi-)simulations bisimulation Chapter choice maps class of small closed under small completes the proof composition consider constructed Corollary denote descent data diagram epimorphism equivalence relation example exponential finite follows forest F free algebra free ZF-algebra freeness functor G-object groupoid height function hence homomorphism of ZF-algebras inductive isomorphism Lemma monad mono monotone successor mutually inverse isomorphisms natural numbers object Neumann ordinals open maps order logic ordinal numbers P-algebra partial order poset power-set axiom preorder presheaf pretopos projective properties Proposition proves quasi-pullback satisfies separation axiom slice category small object small set small suprema small weakly directed small well-founded forest subobject classifier subset successor algebra successor operation sup-lattice supremum surjective Tarski forest Theorem 1.2 transitive sets unique homomorphism unique map universal small map weak algebras weak simulation well-founded small forest Zermelo-Fraenkel algebra Zermelo-Fraenkel set theory