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Gap Forcing: Generalizing the Lévy-Solovay Theorem

Published online by Cambridge University Press:  15 January 2014

Joel David Hamkins
Joel David Hamkins*
Affiliation:
Department of Mathematics, City University of New York, College of Staten Island, 2800 Victory Boulevard, Staten Island, NY 10314, USAE-mail:[email protected]
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Abstract

The Lévy-Solovay Theorem[8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 5 , Issue 2 , June 1999 , pp. 264 - 272
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Hamkins, Joel David, Gap forcing, submitted to the Journal of Mathematical Logic, currently available on the author's web page http://www.math.csi.cuny.edu/~hamkins.Google Scholar
[2] Hamkins, Joel David, Canonical seeds and Prikry trees, this Journal, vol. 62 (1997), no. 2, pp. 373–396.Google Scholar
[3] Hamkins, Joel David, Destruction or preservation as you like it, Annals of Pure and Applied Logic, vol. 91 (1998), pp. 191–229.Google Scholar
[4] Hamkins, Joel David, Small forcing makes any cardinal superdestructible, this Journal, vol. 63 (1998), no. 1, pp. 51–58.Google Scholar
[5] Hamkins, Joel David and Shelah, Saharon, Superdestructibility: a dual to the Laver preparation, this Journal, vol. 63 (1998), no. 2, pp. 549–554.Google Scholar
[6] Hamkins, Joel David and Woodin, W. Hugh, Small forcing creates neither strong nor Woodin cardinals, to appear in the Proceedings of the American Mathematical Society.Google Scholar
[7] Laver, Richard, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385–388.CrossRefGoogle Scholar
[8] Levy, Azriel and Solovay, Robert M., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234–248.CrossRefGoogle Scholar
[9] Scott, Dana S., Measurable cardinals and constructible sets, Bulletin of the Polish Academy of Sciences, Mathematics, vol. 9 (1961), pp. 521–524.Google Scholar
[10] Silver, Jack, The consistency of the generalized continuum hypothesiswith the existence of a measurable cardinal, Axiomatic set theory (Scott, D., editor), vol. I, Proceedings of Symposia in Pure Mathematics, no. 13, American Mathematical Society, 1971, pp. 383–390.Google Scholar