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.


03E25 Axiom of choice and related propositions
Full Text: DOI


[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.