Compactness phenomena in HOD
We prove two compactness theorems for HOD. First, if $\kappa $ is a strong limit singular cardinal with uncountable cofinality and for stationarily many $\delta <\kappa $ , $(\delta ^+)^{\mathrm {HOD}}=\delta ^+$ , then $(\kappa ^+)^{\mathrm {HOD}}=\kappa ^+$ . Second, if...
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| Main Authors: | , |
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| 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/S2050509425100704/type/journal_article |
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| Summary: | We prove two compactness theorems for HOD. First, if
$\kappa $
is a strong limit singular cardinal with uncountable cofinality and for stationarily many
$\delta <\kappa $
,
$(\delta ^+)^{\mathrm {HOD}}=\delta ^+$
, then
$(\kappa ^+)^{\mathrm {HOD}}=\kappa ^+$
. Second, if
$\kappa $
is a singular cardinal with uncountable cofinality and stationarily many
$\delta <\kappa $
are singular in
$\operatorname {\mathrm {HOD}}$
, then
$\kappa $
is singular in
$\operatorname {\mathrm {HOD}}$
. We also discuss the optimality of these results and show that the first theorem does not extend from
$\operatorname {\mathrm {HOD}}$
to other
$\omega $
-club amenable inner models. |
|---|---|
| ISSN: | 2050-5094 |