Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T09:24:08.225Z Has data issue: false hasContentIssue false
  • English
  • Français

WOODIN FOR STRONG COMPACTNESS CARDINALS

Published online by Cambridge University Press:  14 March 2019

STAMATIS DIMOPOULOS
STAMATIS DIMOPOULOS*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF BRISTOL, UNIVERSITY WALK BRISTOL, BS8 1TW, UKE-mail: stamatiosdimopoulos@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Woodin and Vopěnka cardinals are established notions in the large cardinal hierarchy and it is known that Vopěnka cardinals are the Woodin analogue for supercompactness. Here we give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor’s identity crisis theorem for the first strongly compact cardinal.

Keywords

Primary 03E3503E55forcinglarge cardinalsstrongly compact cardinalsWoodin cardinalsVopěnka cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 301 - 319
Copyright
Copyright © The Association for Symbolic Logic 2019 

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

Apter, A. W., Laver indestructibility and the class of compact cardinals, this Journal, vol. 63 (1998), no. 1, pp. 149–157.Google Scholar
Apter, A. W. and Cummings, J., Identity crises and strong compactness, this Journal, vol. 65 (2000), no. 4, pp. 1895–1910.Google Scholar
Apter, A. W. and Cummings, J., Identity crises and strong compactness. II. Strong cardinals. Archive for Mathematical Logic, vol. 40 (2001), no. 1, pp. 25–38.10.1007/s001530050172CrossRefGoogle Scholar
Apter, A. W. and Sargsyan, G., Identity crises and strong compactness. III. Woodin cardinals. Archive for Mathematical Logic, vol. 45 (2006), no. 3, pp. 307–322.10.1007/s00153-005-0316-9CrossRefGoogle Scholar
Brooke-Taylor, A. D., Indestructibility of Vopěnka’s principle. Archive for Mathematical Logic, vol. 50 (2011), no. 5–6, pp. 515–529.10.1007/s00153-011-0228-9CrossRefGoogle Scholar
Cody, B., Easton’s theorem in the presence of Woodin cardinals. Archive for Mathematical Logic, vol. 52 (2013), no. 5–6, pp. 569–591.10.1007/s00153-013-0332-0CrossRefGoogle Scholar
Cummings, J., Iterated forcing and elementary embeddings, Handbook of Set Theory, vol. 1 (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 775–883.Google Scholar
Dimopoulos, S., Woodin cardinals and forcing, preprint, 2017, arXiv:1711.02962.Google Scholar
Hamkins, J. D., Extensions with the approximation and cover properties have no new large cardinals. Fundamenta Mathematicae, vol. 180 (2003), no. 3, pp. 257–277.10.4064/fm180-3-4CrossRefGoogle Scholar
Hamkins, J. D., Forcing and large cardinals, unpublished manuscript.Google Scholar
Jech, T., Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, The third millennium edition, revised and expanded.Google Scholar
Kanamori, A., The Higher Infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2009.Google Scholar
Magidor, M., How large is the first strongly compact cardinal? or A study on identity crises. Annals of Mathematical Logic, vol. 10 (1976), no. 1, pp. 33–57.10.1016/0003-4843(76)90024-3CrossRefGoogle Scholar
Perlmutter, N. L., The large cardinals between supercompact and almost-huge. Archive for Mathematical Logic, vol. 54 (2015), no. 3–4, pp. 257–289.10.1007/s00153-014-0410-yCrossRefGoogle Scholar