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Fragile measurability

  • Joel Hamkins (a1)
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.

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References
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[D] Dodd, A., The core model, Cambridge University Press, London and New York, 1982.
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[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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