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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:
 3e+60 Determinacy principles 3e+35 Consistency and independence results 3e+55 Large cardinals
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##### References:
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