The number of normal measures

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 = κ++, 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 cofinalitypreserving 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. Theorem 1 Assume GCH. Suppose that κ is measurable and let α be a cardinal at most κ. Then in a cofinality-preserving forcing extension, κ carries exactly α normal measures. ∗The first author wishes to thank the Austrian Science Fund (FWF) for its generous support through Project Number P 19375-N18.