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The number of normal measures. (English) Zbl 1183.03042
Without much ado, the authors state and prove their first main result: If GCH holds, \(\kappa\) is measurable and \(\alpha\) is a cardinal not larger than \(\kappa^{++}\) then there is a cofinality-preserving forcing extension in which \(\kappa\) carries exactly \(\alpha\) normal measures. A coding trick applied to generalized Sacks forcing enables the authors to create the desired number of normal ultrafilters.
Similar results are obtained together with the failure of GCH at \(\kappa^+\) or \(\kappa\), e.g., one can have \(2^\kappa=\kappa^{++}\) and exactly \(\alpha\) many measures on \(\kappa\) for any prescribed \(\alpha\leq\kappa^{++}\).
Reviewer: K. P. Hart (Delft)

MSC:
03E55 Large cardinals
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
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References:
[1] Annals of Pure and Applied Logic 65 (1993)
[2] The ordering on normal ultrafilters 51 (1985)
[3] Proceedings of the American Mathematical Society 135 (2007)
[4] Inner models and large cardinals 5 (2002)
[5] Sets constructible from sequences of ultrafilters 39 (1974)
[6] The core model 61 (1982)
[7] Annals of Mathematical Logic 2 pp 359– (1970)
[8] Perfect trees and elementary embeddings 73 pp 906– (2008)
[9] Fine structure and class forcing (2000) · Zbl 0954.03045
[10] Coding over a measurable cardinal 54 pp 1145– (1989)
[11] Annals of Mathematical Logic 1 (1970)
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