A model of the Axiom of Determinacy in which every set of reals is universally Baire

Larson, Paul B.  ; Sargsyan, Grigor  ; Wilson, Trevor 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e94

Résumé

The consistency of the theory $\mathsf {ZF} + \mathsf {AD}_{\mathbb {R}} + {}$‘every set of reals is universally Baire’ is proved relative to $\mathsf {ZFC} + {}$‘there is a cardinal that is a limit of Woodin cardinals and of strong cardinals’. The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf {ZF} + \mathsf {AD}_{\mathbb {R}} + {}$‘every set of reals is Suslin’ is consistent relative to $\mathsf {ZFC} + {}$‘there is a cardinal $\lambda $ that is a limit of Woodin cardinals and of $\mathord {<}\lambda $-strong cardinals’. The $\Sigma ^2_1$ reflection property of our model is proved using genericity iterations as in Neeman [18] and Steel [22].

Détail

DOI : 10.1017/fms.2025.10053

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