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Gap Forcing: Generalizing the Lévy-Solovay Theorem

  • Joel David Hamkins (a1)

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.

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[1] Hamkins, Joel David, Gap forcing, submitted to the Journal of Mathematical Logic, currently available on the author's web page
[2] Hamkins, Joel David, Canonical seeds and Prikry trees, this Journal, vol. 62 (1997), no. 2, pp. 373396.
[3] Hamkins, Joel David, Destruction or preservation as you like it, Annals of Pure and Applied Logic, vol. 91 (1998), pp. 191229.
[4] Hamkins, Joel David, Small forcing makes any cardinal superdestructible, this Journal, vol. 63 (1998), no. 1, pp. 5158.
[5] Hamkins, Joel David and Shelah, Saharon, Superdestructibility: a dual to the Laver preparation, this Journal, vol. 63 (1998), no. 2, pp. 549554.
[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.
[7] Laver, Richard, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385388.
[8] Levy, Azriel and Solovay, Robert M., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234248.
[9] Scott, Dana S., Measurable cardinals and constructible sets, Bulletin of the Polish Academy of Sciences, Mathematics, vol. 9 (1961), pp. 521524.
[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. 383390.
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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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