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Laver indestructibility and the class of compact cardinals

Published online by Cambridge University Press:  12 March 2014


Arthur W. Apter
Affiliation:
Department of Mathematics, Baruch College of Cuny, New York, New York 10010, USA, E-mail: awabb@cunyvm.cuny.edu
Corresponding
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Abstract

Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every κ ∈ K is a supercompact cardinal indestructible under ∈-directed closed forcing, and every κ a measurable limit point of K is a strongly compact cardinal indestructible under κ-directed closed forcing not changing ℘(κ). We then derive as a corollary a model for the existence of a strongly compact cardinal κ which is not κ+ supercompact but which is indestructible under κ-directed closed forcing not changing ℘(κ) and remains non-κ+ supercompact after such a forcing has been done.


Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1998

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References

[1]Apter, A., Some results on consecutive large cardinals, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 117.CrossRefGoogle Scholar
[2]Apter, A., Some results on consecutive large cardinals II: Applications of Radin forcing, Israel Journal of Mathematics, vol. 52 (1985), pp. 273292.CrossRefGoogle Scholar
[3]Apter, A. and Gitik, M., The least measurable can be strongly compact and indestructible, to appear in this Journal.Google Scholar
[4]Apter, A. and Hamkins, J., Weakly indestructible superdestructible compact cardinals, in preparation.Google Scholar
[5]Apter, A. and Shelah, S., Menas' result is best possible, to appear in Transactions of the American Mathematical Society.Google Scholar
[6]Baumgartner, J., Iterated forcing, Surveys in set theory (Mathias, A., editor), Cambridge University Press, Cambridge, England, pp. 159.Google Scholar
[7]Burgess, J., Forcing, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 403452.CrossRefGoogle Scholar
[8]Diprisco, C. and Henle, J., On the compactness of N1 and N2, this Journal, vol. 43 (1978), pp. 394401.Google Scholar
[9]Foreman, M., Magidor, M., and Shelah, S., Martin's maximum, saturated ideals, and non-regular ultrafilters: Part I, Annals of Mathematics, vol. 127 (1988), pp. 147.CrossRefGoogle Scholar
[10]Foreman, M., Magidor, M., and Shelah, S., Martin's maximum, saturated ideals, and non-regular ultrafilters:Part II, Annals of Mathematics, vol. 127 (1988), pp. 521545.CrossRefGoogle Scholar
[11]Gitik, M. and Shelah, S., On certain indestructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic, vol. 28 (1989), pp. 3542.CrossRefGoogle Scholar
[12]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[13]Kanamori, A., The higher infinite, Springer-Verlag, New York and Berlin, 1994.Google Scholar
[14]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Lecture Notes in Mathematics, no. 669, Springer-Verlag, Berlin, 1978, pp. 99275.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. 385388.CrossRefGoogle Scholar
[17]Lévy, A. and Solovay, R., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234248.CrossRefGoogle Scholar
[18]Magidor, M., On the singular cardinals problem II, Annals of Mathematics, vol. 106 (1977), pp. 517547.CrossRefGoogle Scholar
[19]Menas, T., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1975), pp. 327359.CrossRefGoogle Scholar
[20]Shoenfield, J., Unramified forcing, Axiomatic set theory (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, no. 13, American Mathematical Society, Providence, 1971, pp. 357382.CrossRefGoogle Scholar
[21]Solovay, R., Reinhardt, W., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.CrossRefGoogle Scholar
[22]Spector, M., The κ-closed unbounded filter and supercompact cardinals, this Journal, vol. 46 (1981), pp. 31κ40.Google Scholar

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