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  (for α=1),  (for α= κ++, the maximum possible) and  (for α=κ⁺, after collapsing κ++). In addition, under stronger large cardinal hypotheses, one can handle the remaining cases:  (starting with a measurable cardinal of Mitchell order α),  (as in , but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and  (as in , 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 . In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails.
Sy-David Friedman. Menachem Magidor. "The number of normal measures." J. Symbolic Logic 74 (3) 1069 - 1080, September 2009. https://doi.org/10.2178/jsl/1245158100