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Regular ultrafilters and finite square principles

Published online by Cambridge University Press:  12 March 2014

Juliette Kennedy
Affiliation:
Department of Philosophy, Utrecht University, Netherlands Department of Mathematics and Statistics, University of Helsinki, Finland, E-mail: jkennedy@cc.helsinki.fi
Saharon Shelah
Affiliation:
Department of Philosophy, Utrecht University, Netherlands Institute of Mathematics, Hebrew University, Jerusalem, Israel, E-mail: shelah@math.huji.ac.il
Jouko Väänänen
Affiliation:
Department of Philosophy, Utrecht University, Netherlands Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, Netherlands, E-mail: jouko.vaananen@helsinki.fi

Abstract

We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which Mλ/D is not λ++-universal and elementarily equivalent models M and N of size λ for which Mλ/D and Nλ/D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [1].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Chang, C. C. and Keisler, J., Model theory, North-Holland, 1990.Google Scholar
[2]Jensen, R., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308, erratum, R. Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), p. 443, with a section by Jack Silver.CrossRefGoogle Scholar
[3]Kennedy, J. and Sheiah, S., On regular reduced products, this Journal, vol. 67 (2002), no. 3, pp. 11691177.Google Scholar
[4]Kennedy, J. and Sheiah, S., More on regular reduced products, this Journal, vol. 69 (2004), no. 4, pp. 12611266.Google Scholar
[5]Magidor, M. and Shelah, S., The tree property at successors of singular cardinals, Archive for Mathematical Logic, vol. 35 (1996), no. 5-6, pp. 385404.CrossRefGoogle Scholar
[6]Shelah, S., “Gap 1” two-cardinal principles and the omitting types theorem for L(Q), Israel Journal of Mathematics, vol. 65 (1989), no. 2, pp. 133152.CrossRefGoogle Scholar
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