## Algebraic Set TheoryThis 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 |

### Common terms and phrases

Appendix arrow assume axiom belongs bisimulation Chapter choice class of small clearly composition consider constructed Corollary cover defined definition denote descent diagram element equipped example exists exponentiable fact finite fixed follows forest forest F free algebra freeness functor Furthermore geometry given gives groups height function hence holds identity inclusion inductive initial internal isomorphism K-small Lemma means models natural numbers object Note object Observe obtains obvious open maps operation particular poset power-set preserves pretopos projective Proof properties Proposition proves pullback Recall Remark representable satisfies separation set theory simulation small forest small maps small object small set small sups small well-founded forest space square structure subobject successor successor algebra suffices sums suppose suprema supremum Tarski Theorem topos unique Vzee weak weakly directed write Zermelo-Fraenkel ZF-algebra