A model of the Axiom of Determinacy in which every set of reals is universally Baire
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 c...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
|
| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100534/type/journal_article |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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]. |
|---|---|
| ISSN: | 2050-5094 |