Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-12T19:41:20.539Z Has data issue: false hasContentIssue false
  • English
  • Français

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: sdf@logic.univie.ac.at
Menachem Magidor
Affiliation:
Institute of Mathematics, Hebrew University, 91905 Jerusalem, Israel, E-mail: mensara@savion.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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