A note on indestructibility and strong compactness

• Autorzy: Apter A. W.
• Języki publikacji: EN

Abstrakty

EN

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+, ∞)-distributive and λ is 2λ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], {δ < κ | δ is δ+ strongly compact yet δ is not δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is 2δ = δ+ supercompact, κ's supercompactness is indestructible under κ-directed closed forcing which is also (κ+, ∞)-distributive, and for every measurable cardinal δ, δ is δ+ strongly compact if δ is δ+ supercompact.

Słowa kluczowe

EN

supercompact; strongly compact; indestructibility; non-reflecting stationary set of ordinals

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2008
• Tom: Vol. 56, no 3-4
• Strony: 191--197
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Apter A. W.

• Department of Mathematics Baruch College of CUNY New York, NY 10010, U.S.A.,

Bibliografia

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• [3] A. Apter and J. D. Hamkins, Indestructibility and the level-by-level agreement between strong compactness and supercompactness, J. Symbolic Logic 67 (2002), 820-840.
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• [5] J. D. Hamkins, Extensions with the approximation and cover properties have no new large cardinals, Fund. Math. 180 (2003), 257-277.
• [6] —, Gap forcing, Israel J. Math. 125 (2001), 237-252.
• [7] —, Gap forcing: generalizing the Levy-Solovay theorem, Bull. Symbolic Logic 5 (1999), 264-272.
• [8] T. Jech, Set Theory: The Third Millennium Edition, Revised and Expanded, Springer, Berlin, 2003.
• [9] J. Ketonen, Strong compactness and other cardinal sins, Ann. Math. Logic 5 (1972), 47-76.
• [10] R. Laver, Making the supercompactness of K indestructible under K-directed closed forcing, Israel J. Math. 29 (1978), 385-388.
• [11] A. Levy and R. Solovay, Measurable cardinals and the continuum hypothesis, ibid. 5 (1967), 234-248.
• [12] T. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974), 327-359.