Skip to main content Accessibility help

Small forcing makes any cardinal superdestructible

  • Joel David Hamkins (a1)

Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal κ becomes superdestructible—any further <κ-closed forcing which adds a subset to κ will destroy the measurability, even the weak compactness, of κ. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.

Hide All
[1]Apter, Arthur and Shelah, Sarahon, Menas' result is best possible, to appear in Transactions of the American Mathematical Society.
[2]Barbenel, Julius B., Making the hugeness of κ resurrectable after κ-directed closed forcing, Fundamenta Mathematicae, vol. 137 (1991), pp. 924.
[3]Hamkins, Joel David, Fragile measurability, this Journal, vol. 59 (1994), pp. 262282.
[4]Jech, Thomas, Set theory, Academic Press, London, 1978.
[5]Laver, Richard, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385388.
[6]Woodin, W. Hugh, Forcing to a model of a supercompact κ whose weak compactness is killed by Add(κ,1), unpublished theorem.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed