Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-17T16:01:39.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Identity crises and strong compactness

Published online by Cambridge University Press:  12 March 2014

Arthur W. Apter  and
James Cummings
Arthur W. Apter
Affiliation:
Department of Mathematics, Baruch College of Cuny, New York, New York 10010, USA, E-mail:[email protected]
James Cummings
Affiliation:
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA, E-mail:[email protected]. edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals k1..…kn are so that ki; for i = 1..…n is both the ith measurable cardinal and supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.

Keywords

Strongly compact cardinalsupercompact cardinalmeasurable cardinalidentity crisisreverse Easton iteration
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1895 - 1910
Copyright
Copyright © Association for Symbolic Logic 2000

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

[A83]Apter, Arthur W., Some results on consecutive large cardinals, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 1–17.CrossRefGoogle Scholar
[A95]Apter, Arthur W., On the first n strongly compact cardinals, Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 2229–2235.Google Scholar
[A97a]Apter, Arthur W., More on the least strongly compact cardinal, Mathematical Logic Quarterly, vol. 43 (1997), pp. 427–430.CrossRefGoogle Scholar
[A97b]Apter, Arthur W., Patterns of compact cardinals, Annals of Pure and Applied Logic, vol. 89 (1997), pp. 101–115.CrossRefGoogle Scholar
[A98]Apter, Arthur W., Laver indestructibility and the class of compact cardinals, this Journal, vol. 63 (1998), pp. 149–157.Google Scholar
[A99]Apter, Arthur W., On measurable limits of compact cardinals, this Journal, vol. 64 (1999), pp. 1675–1688.Google Scholar
[AS97b]After, Arthur W. and Shelah, Saharon, Menas' result is best possible, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 2007–2034.Google Scholar
[AS97a]Apter, Arthur W., On the strong equality between supercompactness and strong compactness, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 103–128.CrossRefGoogle Scholar
[B]Burgess, J., Forcing, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 403–452.CrossRefGoogle Scholar
[CFM]Cummings, J., Foreman, M., and Magidor, M., Squares, scales, and stationary reflection, to appear in the Journal of Mathematical Logic.Google Scholar
[CW]Cummings, J. and Woodin, W.H., Generalised Prikry forcings, circulated manuscript of a forthcoming book.Google Scholar
[C]Cummings, James, A model in which GCH holds at successors but fails at limits, Transactions of the American Mathematical Society, vol. 329 (1992), pp. 1–39.CrossRefGoogle Scholar
[DH]Di Prisco, C. A. and Henle, J., On the compactness of ℵ1 and ℵ1, this Journal, vol. 43 (1978), pp. 394–401.Google Scholar
[H]Hamkins, J., Lifting and extending measures; fragile measurability, Ph.D. thesis, University of California, Berkeley, 1994.CrossRefGoogle Scholar
[J]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[Ka]Kanamori, A., The higher infinite, Springer-Verlag, Berlin, 1994.Google Scholar
[KaM]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Lecture notes in mathematics 669, Springer-Verlag, Berlin, 1978, pp. 99–275.Google Scholar
[KiM]Kimchi, Y. and Magidor, M., The independence between the concepts of compactness and supercompactness, circulated manuscript.Google Scholar
[L]Laver, Richard, Making the supercompactness of K indestructible under n-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385–388.CrossRefGoogle Scholar
[LS]LÉvy, A. and Solovay, R. M., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234–248.CrossRefGoogle Scholar
[Ma]Magidor, Menachem, How large is the first strongly compact cardinal?, Annals of Mathematical Logic, vol. 10 (1976), pp. 33–57.CrossRefGoogle Scholar
[MS]Mekler, A. and Shelah, S., When does k-free imply strongly k-Free?, Proceedings of the Third Conference on Abelian Group Theory, Gordon and Breach, Salzburg, 1987, pp. 137–148.Google Scholar
[Me]Menas, Telis K., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974), pp. 327–359.CrossRefGoogle Scholar
[SRK]Solovay, Robert M., Reinhardt, William N., and Kanamori, Akihiro, Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.CrossRefGoogle Scholar