Easton's theorem and large cardinals

13 years 5 months ago

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The continuum function F on regular cardinals is known to have great freedom; if , are regular cardinals, then F needs only obey the following two restrictions: (1) cf(F()) > , (2) < F() F(). However, if we wish to preserve measurable cardinals in the generic extension, new restrictions must be put on F. We say that is F()-hypermeasurable if there is an elementary embedding j : V M with critical point such that H(F())V M; j will be called the witnessing embedding. We will show that if , closed under F, is F()-hypermeasurable in V and there is a witnessing embedding j such that j(F)() F(), then will remain measurable in some generic extension realizing F. Key words: Easton's theorem, hypermeasurable and strong cardinals, lifting. AMS subject code classification: 03E35,03E55.

Sy D. Friedman, Radek Honzik

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