Academic Journal
$\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal
| Title: | $\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal |
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| Authors: | Blue, Douglas, Sargsyan, Grigor |
| Contributors: | Narodowe Centrum Nauki |
| Source: | Forum of Mathematics, Sigma ; volume 13 ; ISSN 2050-5094 |
| Publisher Information: | Cambridge University Press (CUP) |
| Publication Year: | 2025 |
| Description: | Following [1], given cardinals $\kappa <\lambda $ , we say $\kappa $ is a club $\lambda $ -Berkeley cardinal if for every transitive set N of size $<\lambda $ such that $\kappa \subseteq N$ , there is a club $C\subseteq \kappa $ with the property that for every $\eta \in C$ , there is an elementary embedding $j: N\rightarrow N$ with $\mathrm {crit }(j)=\eta $ . We say $\kappa $ is $ u $ -club $\lambda $ -Berkeley if $C\subseteq \kappa $ as above is a $ u $ -club. We say $\kappa $ is $\lambda $ -Berkeley if C is unbounded in $\kappa $ . We show that under $\textsf {AD}^{+}$ , (1) every regular Suslin cardinal is $\omega $ -club $\Theta $ -Berkeley (see Theorem 7.1), (2) $\omega _1$ is club $\Theta $ -Berkeley (see Theorem 3.1 and Theorem 7.1), and (3) the ’s are $\Theta $ -Berkeley – in particular, $\omega _2$ is $\Theta $ -Berkeley (see Remark 7.5). Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see Theorem 5.1). This topic has been studied in [31] and [4], albeit from a different point of view. We also show that, assuming $V=L({\mathbb {R}})+{\textsf {AD}}$ , $\omega _1$ is not $\Theta ^+$ -Berkeley, so the result stated in the title is optimal (see Theorem 9.14 and Theorem 9.19). |
| Document Type: | article in journal/newspaper |
| Language: | English |
| DOI: | 10.1017/fms.2025.10082 |
| Availability: | https://doi.org/10.1017/fms.2025.10082 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S2050509425100820 |
| Rights: | https://creativecommons.org/licenses/by/4.0 |
| Accession Number: | edsbas.BC5FF34A |
| Database: | BASE |
| DOI: | 10.1017/fms.2025.10082 |
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