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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:awabb@cunyvm.cuny.edu
James Cummings
Affiliation:
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA, E-mail:jcumming+@andrew.cmu. edu
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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 This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.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

Information

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

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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