Journal of Symbolic Logic

Exactly controlling the non-supercompact strongly compact cardinals

Arthur W. Apter and Joel David Hamkinks

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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 [A97], due to the first author.

Article information

Source
J. Symbolic Logic, Volume 68, Issue 2 (2003), 669- 688.

Dates
First available in Project Euclid: 11 May 2003

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1052669070

Digital Object Identifier
doi:10.2178/jsl/1052669070

Mathematical Reviews number (MathSciNet)
MR1976597

Zentralblatt MATH identifier
1056.03030

Citation

Apter, Arthur W.; Hamkinks, Joel David. Exactly controlling the non-supercompact strongly compact cardinals. J. Symbolic Logic 68 (2003), no. 2, 669-- 688. doi:10.2178/jsl/1052669070. https://projecteuclid.org/euclid.jsl/1052669070


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References

  • Y. Abe Some results concerning strongly compact cardinals, Journal of Symbolic Logic, vol. 50 (1985), pp. 874--880.
  • A. Apter On the least strongly compact cardinal, Israel Journal of Mathematics, vol. 35 (1980), pp. 225--233.
  • A. Apter and J. Cummings Identity crises and strong compactness, Journal of Symbolic Logic, vol. 65 (2000), pp. 1895--1910.
  • A. Apter and M. Gitik The least measurable can be strongly compact and indestructible, Journal of Symbolic Logic, vol. 63 (1998), pp. 1404--1412.
  • A. Apter and J. D. Hamkins Indestructibility and the level-by-level agreement between strong compactness and supercompactness, Journal of Symbolic Logic, vol. 67 (2002), pp. 820--840.
  • A. Apter and S. Shelah On the strong equality between supercompactness and strong compactness, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 103--128.
  • J. Burgess Forcing, Handbook of Mathematical Logic (J. Barwise, editor), North-Holland, Amsterdam,1977, pp. 403--452.
  • J. Cummings A model in which GCH holds at successors but fails at limits, Transactions of the American Mathematical Society, vol. 329 (1992), pp. 1--39.
  • M. Foreman More saturated ideals, Cabal Seminar 79-81, Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, Berlin and New York,1983, pp. 1--27.
  • M. Gitik Changing cofinalities and the nonstationary ideal, Israel Journal of Mathematics, vol. 56 (1986), pp. 280--314.
  • J. D. Hamkins Destruction or preservation as you like it, Annals of Pure and Applied Logic, vol. 91 (1998), pp. 191--229.
  • J. D. Hamkins and W. H. Woodin Small forcing creates neither strong nor Woodin cardinals, Proceedings of the American Mathematical Society, vol. 128 (2000), pp. 3025--3029.
  • A. Kanamori The Higher Infinite, Springer-Verlag, Berlin and New York,1994.
  • Y. Kimchi and M. Magidor The independence between the concepts of compactness and supercompactness, circulated manuscript.
  • R. Laver Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385--388.
  • A. Lévy and R. Solovay Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234--248.
  • M. Magidor How large is the first strongly compact cardinal?, Annals of Mathematical Logic, vol. 10 (1976), pp. 33--57.
  • T. Menas On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974), pp. 327--359.
  • R. Solovay Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics, American Mathematical Society,1974, pp. 365--372.
  • R. Solovay, W. Reinhardt, and A. Kanamori Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73--116.