Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-16T07:28:50.480Z Has data issue: false hasContentIssue false
  • English
  • Français

Small forcing makes any cardinal superdestructible

Published online by Cambridge University Press:  12 March 2014

Joel David Hamkins
Joel David Hamkins*
Affiliation:
Mathematics 15-215, City University of New York, CSI, 2800 Victory Blvd., Staten Island, NY 10314, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 51 - 58
Copyright
Copyright © Association for Symbolic Logic 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Apter, Arthur and Shelah, Sarahon, Menas' result is best possible, to appear in Transactions of the American Mathematical Society.Google Scholar
[2]Barbenel, Julius B., Making the hugeness of κ resurrectable after κ-directed closed forcing, Fundamenta Mathematicae, vol. 137 (1991), pp. 9–24.CrossRefGoogle Scholar
[3]Hamkins, Joel David, Fragile measurability, this Journal, vol. 59 (1994), pp. 262–282.Google Scholar
[4]Jech, Thomas, Set theory, Academic Press, London, 1978.Google Scholar
[5]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
[6]Woodin, W. Hugh, Forcing to a model of a supercompact κ whose weak compactness is killed by Add(κ,1), unpublished theorem.Google Scholar