Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T10:29:52.881Z Has data issue: false hasContentIssue false
  • English
  • Français

Supercompactness and measurable limits of strong cardinals

Published online by Cambridge University Press:  12 March 2014

Arthur W. Apter
Arthur W. Apter*
Affiliation:
Department of Mathematics, Baruch College of Cuny, New York NY 10010, USA, E-mail: awabb@;cunyvm.cuny.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, two theorems concerning measurable limits of strong cardinals and supercompactness are proven. This generalizes earlier work, both individual and joint with Shelah.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 629 - 639
Copyright
Copyright © Association for Symbolic Logic 2001

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, A., Some new upper bounds in consistency strength for certain choiceless large cardinal patterns, Archive for Mathematical Logic, vol. 31 (1992), pp. 201–205.CrossRefGoogle Scholar
[2]Apter, A., Laver indestructibility and the class of compact cardinals, this Journal, vol. 63 (1998), pp. 149–157.Google Scholar
[3]Apter, A., On measurable limits of compact cardinals, this Journal, vol. 64 (1999), pp. 1675–1688.Google Scholar
[4]Apter, A. and Cummings, J., Identity crises and strong compactness II: strong cardinals, Archive for Mathematical Logic, to appear.Google Scholar
[5]Apter, A. and Shelah, S., Menas’ result is best possible, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 2007–2034.CrossRefGoogle Scholar
[6]Apter, A. and Shelah, S., On the strong equality between supercompactness and strong compactness, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 103–128.CrossRefGoogle Scholar
[7]Burgess, J., Forcing, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 403–452.CrossRefGoogle Scholar
[8]Cummings, J., 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
[9]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
[10]Hamkins, J., Gap forcing, to appear in the Israel Iournal of Mathematics.Google Scholar
[11]Hamkins, J., Destruction or preservation as you like it, Annals of Pure and Applied Logic, vol. 91 (1998), pp. 191–229.CrossRefGoogle Scholar
[12]Hamkins, J., Gap forcing: generalizing the Lévy-Solovay Theorem, The Bulletin of Symbolic Logic, vol. 5 (1999), pp. 264–272.CrossRefGoogle Scholar
[13]Hamkins, J. and Woodin, H., Small forcing and strong cardinals, Proceedings of the American Mathematical Society, vol. 128 (2000), pp. 3025–3029.CrossRefGoogle Scholar
[14]Kanamori, A., The Higher Infinite, Springer-Verlag, Berlin and Heidelberg, 1994.Google Scholar
[15]Kimchi, Y. and Magidor, M., The independence between the concepts of compactness and supercompactness, circulated manuscript.Google Scholar
[16]Laver, R., Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978). pp. 385–388.CrossRefGoogle Scholar
[17]Lévy, A. and Solovay, R. M., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234–248.CrossRefGoogle Scholar
[18]Magidor, M., Changing cofinality of cardinals, Fundamenta Mathematicae, vol. 99 (1978). pp. 61–71.CrossRefGoogle Scholar
[19]Mekler, A. and Shelah, S., When κ-free implies strongly κ-free, Proceedings of the Third Conference on Ahelian group theory, Gordon and Breach, Salzburg, 1987, pp. 137–148.Google Scholar
[20]Menas, T., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974), pp. 327–359.CrossRefGoogle Scholar
[21]Radin, L., Adding closed cofinal sequences to large cardinals, Annals of Mathematical Logic, vol. 22 (1982), pp. 243–261.CrossRefGoogle Scholar