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Fragile measurability

Published online by Cambridge University Press:  12 March 2014

Joel Hamkins
Joel Hamkins*
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, E-mail: HAMKINS@MATH.BERKELEY.EDU
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Abstract

Laver [L] and others [G-S] have shown how to make the supercompactness or strongness of κ indestructible by a wide class of forcing notions. We show, alternatively, how to make these properties fragile. Specifically, we prove that it is relatively consistent that any forcing which preserves κ<κ and κ+, but not P(κ), destroys the measurability of κ, even if κ is initially supercompact, strong, or if I1(κ) holds. Obtained as an application of some general lifting theorems, this result is an “inner model” type of theorem proved instead by forcing.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 1 , March 1994 , pp. 262 - 282
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

[C-W] Cummings, James and Woodin, Hugh, Generalized Prikry forcings, 1990, preprint.Google Scholar
[D] Dodd, A., The core model, Cambridge University Press, London and New York, 1982.CrossRefGoogle Scholar
[G-S] Gitik, Moti and Shelah, Saharon, On certain indestructability of strong cardinals…, rArchive for Mathematical Logic, vol. 28 (1989), pp. 35–42.CrossRefGoogle Scholar
[J] Jech, Thomas, Set theory, Academic Press, New York, 1978.Google Scholar
[Ka] Kanamori, Akihiro, Large cardinals in set theory (to appear).Google Scholar
[Ku] Künen, Kenneth, Elementary embeddings and infinitary combinatorics, this Journal, vol. 36 (1971), pp. 407–413.Google Scholar
[L] 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
[Me] Menas, T., Consistency results concerning supercompactness, Transactions of the American Mathematical Society, vol. 223 (1976), pp. 61–91.CrossRefGoogle Scholar
[Mi] Mitchell, , Hypermeasurable cardinals, Logic Colloquium '78 (Boffa, et al., editors), North-Holland, Amsterdam, 1978, pp. 303–316.Google Scholar
[S] Silver, J., The consistency of the GCH with the existence of a measurable cardinal, Axiomatic set theory (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 383–390.CrossRefGoogle Scholar
[W1] Woodin, Hugh, Adding a subset to κ may destroy the weak compactness of κ, even if κ is supercompact, personal communication.Google Scholar
[W2] Woodin, Hugh, Woodin cardinals & the stationary tower forcing, notes.Google Scholar
[W3] Woodin, Hugh, Forcing GCH with a strong cardinal, personal communication.Google Scholar