Description:
Results of Gitik
Content:
The following results are from Gitik [1980]. Let \(\Omega = \{\alpha\in\hbox{ On}:
(\forall\beta\le\alpha)(\beta=0\vee(\exists\gamma)(\beta=\gamma+1)\vee(\forall n\in\omega)(\exists\gamma_n\le\beta)(\beta=\bigcup_{n\in\omega}\gamma_n)\}\).
Gitik shows, by making small changes in \(\cal M17\), that each of the following are consistent:
-
\(ZF + \Omega =\) On.
-
\(ZF + \Omega = \omega_{\Omega}\).
-
\(ZF + \Omega = \omega_{\alpha+1}\), for any \(\alpha\).
In fact, item 1 holds in
\(\cal M17\) itself. It is clear that \(ZFC \vdash\Omega = \omega_1\) (
Form 315).
In Gitik [1985], the following theorem is given:
Suppose \(M\) is a countable model of \(ZFC\) and \(\kappa\) is an almost huge cardinal in \(M\). (See Note 20
for the definition of various types of inaccessible cardinals.) If \(S\) is any subset of \(\kappa\) consisting of non-limit cardinals,
then there is a model \(\cal M(S)\) of \(ZF\) such that the regular cardinals are exactly \(\{\aleph_{\alpha}: \alpha\in S\cup \{0\}\}\).
In the case that \(S=\emptyset\), then all uncountable \(\aleph_{\alpha}\)'s are singular. Note, however, that in \(\cal M17\) all uncountable
\(\aleph_\alpha\)'s are singular. Further, the construction of \(\cal M17\) requires only the existence of a strongly compact cardinal which is
weaker than the existence of an almost huge cardinal.
Howard-Rubin number:
55
Type:
Results
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