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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: sdf@logic.univie.ac.at
Menachem Magidor
Affiliation:
Institute of Mathematics, Hebrew University, 91905 Jerusalem, Israel, E-mail: mensara@savion.huji.ac.il
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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

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