Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T14:32:12.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Strongly unfoldable cardinals made indestructible

Published online by Cambridge University Press:  12 March 2014

Thomas A. Johnstone
Thomas A. Johnstone*
Affiliation:
Mathematics New York City College of Technology, 300 Jay Street, Brooklyn, NY 11201, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

I provide indestructibility results for large cardinals consistent with V = L, such as weakly compact, indescribable and strongly unfoldable cardinals. The Main Theorem shows that any strongly unfoldable cardinal κ can be made indestructible by <κ-closed, κ-proper forcing. This class of posets includes for instance all <κ-closed posets that are either κ−-c.c. or <κ-strategically closed as well as finite iterations of such posets. Since strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, the Main Theorem therefore makes these two large cardinal notions similarly indestructible. Finally. I apply the Main Theorem to obtain a class forcing extension preserving all strongly unfoldable cardinals in which every strongly unfoldable cardinal κ is indestructible by <κ-closed, κ-proper forcing.

Keywords

strongly unfoldable cardinalforcingindestructibility
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1215 - 1248
Copyright
Copyright © Association for Symbolic Logic 2008

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

[CW]Cummings, J. and Woodin, W. H., Generalized Prikry forcings, unpublished manuscript.Google Scholar
[Dev84]Devlin, Keith J., Constructibility, Springer Verlag, Berlin Heidelberg, 1984.CrossRefGoogle Scholar
[DH06]Džamonja, M. and Hamkins, J. D., Diamond (on the regulars) can failat any stronglyunfoldable cardinal, Annals of Pure and Applied Logic, vol. 144 (2006), pp. 83–95.CrossRefGoogle Scholar
[Eis03]Eisworth, Todd, On iterated forcing for successors of regular cardinals, Fundamenta Mathematicae, vol. 179 (2003), pp. 249–266.CrossRefGoogle Scholar
[GS89]Gitik, M. and Shelah, S., On certain indestructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic, vol. 28 (1989), pp. 35–42.CrossRefGoogle Scholar
[HJ]Hamkins, J. D. and Johnstone, T. A., Indestructible strong unfoldability, submitted.Google Scholar
[Ham98]Hamkins, Joel D., Small forcing makes any cardinal superdestructible, this Journal, vol. 63 (1998). no. 1, pp. 51–58.Google Scholar
[Ham00]Hamkins, Joel D., The lottery preparation, Annals of Pure and Applied Logic, vol. 101 (2000), pp. 103–146.CrossRefGoogle Scholar
[Ham01]Hamkins, Joel D., Unfoldable cardinals and the GCH, this Journal, vol. 66 (2001), no. 3, pp. 1186–1198.Google Scholar
[Ham03]Hamkins, Joel D., Extensions with the approximation and cover properties have no new large cardinals, Fundamenta Mathematicae, vol. 180 (2003), no. 3, pp. 257–277.CrossRefGoogle Scholar
[Ham]Hamkins, Joel D.. Lectures on forcing and large cardinals, in preparation.Google Scholar
[Hau91]Hauser, Kai, Indescribable cardinals and elementary embeddings, this Journal, vol. 56 (1991), no. 2, pp. 439–457.Google Scholar
[HR01]Hyttinen, T. and Rautila, M., The canary tree revisited, this Journal, vol. 66 (2001), no. 4, pp. 1677–1694.Google Scholar
[Jec03]Jech, Thomas, Set theory, Springer-Verlag, Berlin Heidelberg New York, 2003.Google Scholar
[KY]KÖNig, B. and Yoshinobu, Y., Kurepa-trees and namba forcing, 2006, arXiv:math.L0/0605130.Google Scholar
[Kun99]Kunen, Kenneth, Set theory, An introduction to independence proofs, North Holland, 1980.Google Scholar
[Lav78]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
[Miy98]Miyamoto, Tadatoshi, A note on weak segments of PFA, Proceedings of the sixth Asian logic conference (River Edge, NJ), World Science Publishers, 1998, pp. 175–197.CrossRefGoogle Scholar
[Rei06]Reitz, Jonas, The ground axiom, Ph.D. Dissertation, The Graduate Center of The City University of New York, 2006.Google Scholar
[Ros]Roslanowski, Andrzej, Shelah's search for properness for iterations with uncountable supports, unpublished talk at MAMLS, Rutgers University, 10 2005.Google Scholar
[RS]Roslanowski, Andrzej and Shelah, Saharon, Sheva-Sheva-Sheva: Large creatures, arXiv:math.L0/ 0210206,2002.Google Scholar
[She80]Shelah, Saharon, Independence results, this Journal, vol. 45 (1980), no. 3, pp. 563–573.Google Scholar
[She98]Shelah, Saharon, Proper and improper forcing, Springer-Verlag, Berlin Heidelberg New York, 1998.CrossRefGoogle Scholar
[Vil98]Villaveces, Andres, Chains of end elementary extensions of models of set theory, this Journal, vol. 63 (1998), no. 3, pp. 1116–1136.Google Scholar