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Exactly controlling the non-supercompact strongly compact cardinals

Published online by Cambridge University Press:  12 March 2014

Arthur W. Apter  and
Joel David Hamkins
Arthur W. Apter
Affiliation:
Department of Mathematics, Baruch College of Cuny, New York, New York 10010, USA, E-mail: [email protected], URL: http://math.baruch.cuny.edu/~apter
Joel David Hamkins
Affiliation:
Department of Mathematics, The College of Staten Island of Cuny Department of Mathematics, The Graduate Center of Cuny, 365 Fifth Avenue, New York, New York 10016, USA, E-mail: [email protected], URL: http://jdh.hamkins.org
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Abstract

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify Theorems 1 and 2 of [5], due to the first author.

Keywords

Supercompact cardinalstrongly compact cardinalstrong cardinalindestructibilitynon-reflecting stationary set of ordinals
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 669 - 688
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1]Abe, Y., Some results concerning strongly compact cardinals, this Journal, vol. 50 (1985), pp. 874–880.Google Scholar
[2]Apter, A., On the least strongly compact cardinal, Israel Journal of Mathematics, vol. 35 (1980), pp. 225–233.CrossRefGoogle Scholar
[3]Apter, A., On the first n strongly compact cardinals, Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 2229–2235.Google Scholar
[4]Apter, A., More on the least strongly compact cardinal, Mathematical Logic Quarterly, vol. 43 (1997), pp. 427–430.CrossRefGoogle Scholar
[5]Apter, A., Patterns of compact cardinals, Annals of Pure and Applied Logic, vol. 89 (1997), pp. 101–115.CrossRefGoogle Scholar
[6]Apter, A., Laver indestructibility and the class of compact cardinals, this Journal, vol. 63 (1998), pp. 149–157.Google Scholar
[7]Apter, A., Strong compactness, measurability, and the class of supercompact cardinals, Fundamenta Mathematicae, vol. 167 (2001), pp. 65–78.CrossRefGoogle Scholar
[8]Apter, A. and Cummings, J., Identity crises and strong compactness, this Journal, vol. 65 (2000), pp. 1895–1910.Google Scholar
[9]Apter, A., Identity crises and strong compactness II: Strong cardinals, Archive for Mathematical Logic, vol. 40 (2001), pp. 25–38.CrossRefGoogle Scholar
[10]Apter, A. and Gitik, M., The least measurable can be strongly compact and indestructible, this Journal, vol. 63 (1998), pp. 1404–1412.Google Scholar
[11]Apter, A. and Hamkins, J. D., Indestructibility and the level-by-level agreement between strong compactness and supercompactness, this Journal, vol. 67 (2002), pp. 820–840.Google Scholar
[12]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
[13]Burgess, J., Forcing, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 403–452.CrossRefGoogle Scholar
[14]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
[15]Foreman, M., More saturated ideals, Cabal Seminar 79-81, Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, Berlin and New York, 1983, pp. 1–27.CrossRefGoogle Scholar
[16]Gitik, M., Changing cofinalities and the nonstationary ideal, Israel Journal of Mathematics, vol. 56 (1986), pp. 280–314.CrossRefGoogle Scholar
[17]Hamkins, J. D., Destruction orpreservation as you like it, Annals of Pure and Applied Logic, vol. 91 (1998), pp. 191–229.CrossRefGoogle Scholar
[18]Hamkins, J. D., Gap forcing: Generalizing the LÉvy-Solovay Theorem, The Bulletin of Symbolic Logic, vol. 5 (1999), pp. 264–272.CrossRefGoogle Scholar
[19]Hamkins, J. D., The lottery preparation, Annals of Pure and Applied Logic, vol. 101 (2000), pp. 103–146.CrossRefGoogle Scholar
[20]Hamkins, J. D., Gap forcing, Israel Journal of Mathematics, vol. 125 (2001), pp. 237–252.CrossRefGoogle Scholar
[21]Hamkins, J. D. and Woodin, W. H., Small forcing creates neither strong nor Woodin cardinals, Proceedings of the American Mathematical Society, vol. 128 (2000), pp. 3025–3029.CrossRefGoogle Scholar
[22]Kanamori, A., The Higher Infinite, Springer-Verlag, Berlin and New York, 1994.Google Scholar
[23]Kimchi, Y. and Magidor, M., The independence between the concepts of compactness and super-compactness, circulated manuscript.Google Scholar
[24]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
[25]Lévy, A. and Solovay, R., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234–248.CrossRefGoogle Scholar
[26]Magidor, M., How large is the first strongly compact cardinal?, Annals of Mathematical Logic, vol. 10 (1976), pp. 33–57.CrossRefGoogle Scholar
[27]Menas, T., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974), pp. 327–359.CrossRefGoogle Scholar
[28]Solovay, R., Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 1974, pp. 365–372.CrossRefGoogle Scholar
[29]Solovay, R., Reinhardt, W., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.CrossRefGoogle Scholar