## 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.

### MSC:

 3e+25 Axiom of choice and related propositions

### Keywords:

illfounded sets; ZF; permutation model
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### 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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