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- Title
COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION.
- Authors
CHAN, WILLIAM; JACKSON, STEPHEN; TRANG, NAM
- Abstract
Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $ , $\vee $ , $\forall ^{\mathbb {R}}$ , continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma)$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $. Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $. Then the countable length everywhere club uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{ with the property that for all $\ell \in {}^{ and clubs $C \subseteq D \subseteq \kappa $ , $R(\ell ,D)$ implies $R(\ell ,C)$ , there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$ , $R(\ell ,\Lambda (\ell))$. In particular, under these assumptions, for all $n \in \omega $ , $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization.
- Subjects
CLUBS; COLLECTIONS
- Publication
Journal of Symbolic Logic, 2023, Vol 88, Issue 4, p1556
- ISSN
0022-4812
- Publication type
Article
- DOI
10.1017/jsl.2022.78