$\textsf {AD}^{+}$ 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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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100820/type/journal_article |
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| author | Douglas Blue Grigor Sargsyan |
| author_facet | Douglas Blue Grigor Sargsyan |
| author_sort | Douglas Blue |
| collection | DOAJ |
| 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
$\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). |
| format | Article |
| id | doaj-art-ebde432cf09242559d078db52cbcc7b9 |
| institution | DOAJ |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-ebde432cf09242559d078db52cbcc7b92025-08-20T02:47:58ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.10082$\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinalDouglas Blue0https://orcid.org/0000-0003-1217-9215Grigor Sargsyan1https://orcid.org/0000-0002-6095-1997Department of Philosophy, https://ror.org/01an3r305University of Pittsburgh, 1017 Cathedral of Learning, Pittsburgh, PA, 15260, USAhttps://ror.org/04hrrh248Institute of Mathematics of Polish Academy of Sciences, ul. Śniadeckich 8, Warszawa, 00-656, Poland; E-mail: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).https://www.cambridge.org/core/product/identifier/S2050509425100820/type/journal_article03E6003E5503E4503E57 |
| spellingShingle | Douglas Blue Grigor Sargsyan $\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal Forum of Mathematics, Sigma 03E60 03E55 03E45 03E57 |
| title | $\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal |
| title_full | $\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal |
| title_fullStr | $\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal |
| title_full_unstemmed | $\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal |
| title_short | $\textsf {AD}^{+}$ implies $ \omega _{1}$ is a club $ \Theta $ -Berkeley cardinal |
| title_sort | textsf ad implies omega 1 is a club theta berkeley cardinal |
| topic | 03E60 03E55 03E45 03E57 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425100820/type/journal_article |
| work_keys_str_mv | AT douglasblue textsfadimpliesomega1isaclubthetaberkeleycardinal AT grigorsargsyan textsfadimpliesomega1isaclubthetaberkeleycardinal |