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On a problem inspired by determinacy. (English) Zbl 0661.03040
The early development of consequences of the axiom of determinateness, AD, led to the rather striking results that \(\aleph_ 1\) and \(\aleph_ 2\) are measurable whereas the \(\aleph_ n\) for \(n>2\) are singular. The question whether all regular cardinals are measurable, however, cannot be solved by AD alone. Building on previous work (with J. Henle) and Gitik’s model of ZF \(+ \) “all uncountable cardinals are singular” the author shows that the above question has a non-trivial positive answer, more precisely, if ZFC \(+ \) “there is an almost huge cardinal” is consistent then so is the theory ZF \(+ \) \(``\aleph_ 1\) is measurable via the club filter” \(+ \) “all uncountable cardinals \(>\) \(\aleph_ 1\) are singular.”
Reviewer: K.Gloede

MSC:
03E60 Determinacy principles
03E35 Consistency and independence results
03E55 Large cardinals
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References:
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