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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 $$R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$$ 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.
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THE TREE PROPERTY AT THE TWO IMMEDIATE SUCCESSORS OF A SINGULAR CARDINAL
Abstract We present an alternative proof that from large cardinals, we can force the tree property at $$\kappa ^+$$ and $$\kappa ^{++}$$ simultaneously for a singular strong limit cardinal $$\kappa $$ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $$\kappa =\aleph _{\omega ^2}$$ .
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- PAR ID:
- 10333323
- Date Published:
- Journal Name:
- The Journal of Symbolic Logic
- Volume:
- 86
- Issue:
- 2
- ISSN:
- 0022-4812
- Page Range / eLocation ID:
- 600 to 608
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jsl.2020.11
