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An application of ultrapowers to changing cofinality

Published online by Cambridge University Press:  12 March 2014

Patrick Dehornoy
Patrick Dehornoy*
Affiliation:
UER de Mathématiques, Université Paris VII, 2 Place Jussieu, 75221 Paris-Cedex 05, France
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Extract

The problem of changing the cofinality of a measurable cardinal to ω with the help of an iterated ultrapower construction has been introduced in [Bu] and more completely studied in [De]. The aim of this paper is to investigate how the construction above has to be changed to obtain an uncountable cofinality for the (previously) measurable cardinal.

A forcing approach of this question has been developed by Magidor in [Ma]. Just as in the countable case with Prikry forcing, it turns out that the needed hypothesis and the models constructed are the same in both techniques. However the ultrapowers yield a solution which may appear as more effective. In particular the sequence used to change the measurable cardinal into a cardinal of cofinality α has the property that for any β < α the restriction to β of this sequence can be used to change the cofinality of the (same) measurable cardinal to β.

The result we prove is as follows:

Theorem. Assume that α is a limit ordinal, that (Uβ)β<α is a sequence of complete ultrafilters on κ > α in the model N0, andfor B included in α let NB be the ultrapower of N0 by those Uβ which are such that β is in B.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 2 , June 1983 , pp. 225 - 235
Copyright
Copyright © Association for Symbolic Logic 1983

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References

REFERENCES

[Bu]Bukovsky, L., Changing cofinality of a measurable cardinal, Commentationes Mathematicae Universitatis Carolinae, vol. 14 (1973), pp. 689–697.Google Scholar
[De]Dehornoy, P., Iterated ultrapowers and Prikry's forcing, Annals of Mathematical Logic, vol. 15 (1978), pp. 109–161.CrossRefGoogle Scholar
[Ma]Magidor, M., Changing cofinalityof cardinals, Fundamenta Mathematicae, vol. 99 (1978), pp. 61–71,CrossRefGoogle Scholar
[Mi]Mitchell, W., Sets constructive fromasequence of ultrafilters, this Journal, vol. 39 (1974), pp. 57–66.Google Scholar
[SRK]Solovay, R., Reinhardt, W. and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1977), pp. 73–116.CrossRefGoogle Scholar