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Some results on consecutive large cardinals. II: Applications of Radin forcing. (English) Zbl 0603.03016
[For Part I see Ann. Pure Appl. Logic 25, 1-17 (1983; Zbl 0548.03030).]
Let \(\kappa\) be a 3 huge cardinal in a countable model V of ZFC, and let A and B be subsets of the successor ordinals \(<\kappa\) so that \(A\cup B=\{\alpha <\kappa:\alpha\) is a successor ordinal\(\}\). Using techniques of Gitik, we construct a choiceless model \(N_ A\) of ZF of height \(\kappa\) so that \(N_ A\vDash ''ZF + \neg AC_{\omega} + For\alpha\in A\), \(\aleph_{\alpha}\) is a Ramsey cardinal \(+ For\beta\in B\), \(\aleph_{\beta}\) is a singular Rowbottom cardinal which carries a Rowbottom filter \(+ For\gamma\) a limit ordinal, \(\aleph_{\gamma}\) is a Jonsson cardinal which carries a Jonsson filter”.

MSC:
03E55 Large cardinals
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