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Fragile measurability

  • Joel Hamkins (a1)

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.

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[D] Dodd, A., The core model, Cambridge University Press, London and New York, 1982.
[G-S] Gitik, Moti and Shelah, Saharon, On certain indestructability of strong cardinals…, rArchive for Mathematical Logic, vol. 28 (1989), pp. 3542.
[J] Jech, Thomas, Set theory, Academic Press, New York, 1978.
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[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. 383390.
[W1] Woodin, Hugh, Adding a subset to κ may destroy the weak compactness of κ, even if κ is supercompact, personal communication.
[W2] Woodin, Hugh, Woodin cardinals & the stationary tower forcing, notes.
[W3] Woodin, Hugh, Forcing GCH with a strong cardinal, personal communication.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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