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The number of normal measures

Published online by Cambridge University Press:  12 March 2014

Sy-David Friedmanc  and
Menachem Magidor
Sy-David Friedmanc
Affiliation:
Kurt Gödel Research Center, University of Vienna, Währinger Strasse 25, A-1090 Vienna, Austria, E-mail: [email protected]
Menachem Magidor
Affiliation:
Institute of Mathematics, Hebrew University, 91905 Jerusalem, Israel, E-mail: [email protected]
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 1069 - 1080
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Apter, A., Cummings, J., and Hamkins, J., Large cardinals with few measures, Proceedings of the American Mathematical Society, vol. 135 (2007), no. 7.CrossRefGoogle Scholar
[2]Baldwin, S., The ⊲-ordering on normal ultrafilters, this Journal, vol. 51 (1985), no. 4.Google Scholar
[3]Ben-Neria, O., Magidor forcing and the number of normal measures, Master's thesis, Tel Aviv University, 2008.Google Scholar
[4]Cummings, J., Possible behaviours for the Mitchell ordering, Annals of Pure and Applied Logic, vol. 65 (1993).CrossRefGoogle Scholar
[5]Dodd, A., The core model, London Math Society Lecture Notes, vol. 61, 1982.CrossRefGoogle Scholar
[6]Friedman, S., Coding over a measurable cardinal, this Journal, vol. 54 (1989), no. 4, pp. 1145–1159.Google Scholar
[7]Friedman, S., Fine structure and class forcing, de Gruyter, 2000.CrossRefGoogle Scholar
[8]Friedman, S. and Thompson, K., Perfect trees and elementary embeddings, this Journal, vol. 73 (2008), no. 3, pp. 906–918.Google Scholar
[9]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970).CrossRefGoogle Scholar
[10]Kunen, K. and Paris, J., Boolean extensions and measurable cardinals, Annals of Mathematical Logic, vol. 2 (1970/1971), no. 4, pp. 359–377.CrossRefGoogle Scholar
[11]Leaning, J., Disassociated indiscernibles, Ph.D. thesis, University of Florida, 1999, (also see [3]).Google Scholar
[12]Mitchell, W., Sets constructible from sequences of ultrafilters, this Journal, vol. 39 (1974), no. 1.Google Scholar
[13]Zeman, M., Inner models and large cardinals, de Gruyter Series in Logic and Its Applications, vol. 5, 2002.CrossRefGoogle Scholar