The number of normal measures

E-ISSN： 1943-5886|74|3|1069-1080

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.74, Iss.3, 2009-09, pp. : 1069-1080

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Abstract

There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH. where α is a cardinal at most κ ⁺⁺. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = α ⁺⁺, the maximum possible) and [1] (for α = κ ⁺, after collapsing κ ⁺⁺). In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of Mitchell order α), [2] (as in [12], but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and [11] (as in [12], but where κ is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing. The proof uses κ-Sacks forcing and the “tuning fork” technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails.