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**ZF + “every set is the same size as a wellfounded set”.**
*(English)*
Zbl 1044.03037

ZF is Zermelo-Fraenkel set theory with the axiom of foundation but without the axiom of choice. ZFB is ZF with the axiom that says that every set is the same size as a wellfounded set. ZFAFA is ZF with foundation replaced by the Forti-Honsell antifoundation axiom, which says that every accessible pointed digraph is the \(\in\)-picture of a unique set. The author proves that: every sentence true in every permutation model of a model of ZF is a theorem of ZFB; ZF and ZFAFA are both extensions of ZFB, conservative for stratified formulæ; the class of models of ZFB is closed under creation of Rieger-Bernays permutation models. This paper is a continuation of the work of Jean Coret on ZF and illfounded sets.

Reviewer: Adonai S. Sant’Anna (Columbia)

### MSC:

03E25 | Axiom of choice and related propositions |

Full Text:
DOI

### References:

[1] | Annals of Pure and Aplied Logic pp 107– (2001) |

[2] | Bulletin de la Société Mathématique de Belgique série B 36 pp 69– (1984) |

[3] | Comptes Rendus hebdomadaires ties séances de l’Académie des Sciences de Paris série A 264 pp 837– (1964) |

[4] | DOI: 10.1002/malq.19900360504 · Zbl 0717.03019 |

[5] | Logic, induction and sets (2003) |

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