$\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal
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...
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| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100820/type/journal_article |
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| Summary: | 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
$\nu $
-club
$\lambda $
-Berkeley if
$C\subseteq \kappa $
as above is a
$\nu $
-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). |
|---|---|
| ISSN: | 2050-5094 |