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The number of normal measures

  • Sy-David Friedmanc (a1) and Menachem Magidor (a2)

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.

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References

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[1]Apter, A., Cummings, J., and Hamkins, J., Large cardinals with few measures, Proceedings of the American Mathematical Society, vol. 135 (2007), no. 7.
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[5]Dodd, A., The core model, London Math Society Lecture Notes, vol. 61, 1982.
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[7]Friedman, S., Fine structure and class forcing, de Gruyter, 2000.
[8]Friedman, S. and Thompson, K., Perfect trees and elementary embeddings, this Journal, vol. 73 (2008), no. 3, pp. 906918.
[9]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970).
[10]Kunen, K. and Paris, J., Boolean extensions and measurable cardinals, Annals of Mathematical Logic, vol. 2 (1970/1971), no. 4, pp. 359377.
[11]Leaning, J., Disassociated indiscernibles, Ph.D. thesis, University of Florida, 1999, (also see [3]).
[12]Mitchell, W., Sets constructible from sequences of ultrafilters, this Journal, vol. 39 (1974), no. 1.
[13]Zeman, M., Inner models and large cardinals, de Gruyter Series in Logic and Its Applications, vol. 5, 2002.

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