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COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 88 / Issue 4 / December 2023
- Published online by Cambridge University Press:
- 21 November 2022, pp. 1556-1572
- Print publication:
- December 2023
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- Article
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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 
$R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all 
$\ell \in {}^{<{\omega _1}}\kappa $ 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.
