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Indestructibility and the level-by-level agreement between strong compactness and supercompactness

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 the City University of New York, New York, NY 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 the City University of New York, 2800 Victory Boulevard, Staten Island, New York 10314. USA The Graduate Center of the City University of New York, 365 Fifth Avenue, New York, New York 10016. USA Department of Mathematical Sciences, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, USA, E-mail: [email protected], URL: http://jdh.hamkins.org
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Abstract

Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets. or less level-by-level agreement, such as requiring it only on measure one sets, then yes. it can.

Keywords

Supercompact cardinalstrongly compact cardinalindestructibilitylevel-by-level agreement
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 2 , June 2002 , pp. 820 - 840
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[Apt]Apter, Arthur W., Some remarks on indestructibility and Hamkins' lottery preparation, Archive for Mathematical Logic, submitted for publication.Google Scholar
[Apt01]Apter, Arthur W., Strong compactness, measurability, and the class of supercompact cardinals, Fundamenta Mathematicae, vol. 167 (2001), pp. 65–78.CrossRefGoogle Scholar
[AC00]Apter, Arthur W. and Cummings, James, Identity crises and strong compactness, this Journal, vol. 65 (2000), no. 4, pp. 1895–1910.Google Scholar
[AC01]Apter, Arthur W. and Cummings, James, Identity crises and strong compactness II: strong cardinals, Archive for Mathematical Logic, vol. 40 (2001), pp. 25–38.CrossRefGoogle Scholar
[AH99]Apter, Arthur W. and Hamkins, Joel David, Universal indestructibility, Kobe Journal of Mathematics, vol. 16 (1999), no. 2, pp. 119–130.Google Scholar
[AS97]Apter, Arthur W. and Shelah, Saharon, On the strong equality between supercompactness and strong compactness, Transactions of the American Mathematical Society, vol. 349 (1997), no. 1, pp. 103–128.CrossRefGoogle Scholar
[Cum92]Cummings, James, A model in which GCH holds at successors but fails at limits, Transactions of the American Mathematical Society, vol. 329 (1992), no. 1, pp. 1–39.CrossRefGoogle Scholar
[For83]Foreman, Matthew, More saturated ideals, Cabal seminar 79–81, Lecture Notes in Mathematics 1019, Springer, Berlin, 1983, pp. 1–27.Google Scholar
[GS89]Gitik, Moti and Shelah, Saharon, On certain indestructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic, vol. 28 (1989), no. 1, pp. 35–42.CrossRefGoogle Scholar
[Ham94]Gitik, Moti and Shelah, Saharon, Fragile measurability, this Journal, vol. 59 (1994), no. 1, pp. 262–282.Google Scholar
[Ham97]Gitik, Moti and Shelah, Saharon, Canonical seeds and Prikry trees, this Journal, vol. 62 (1997), no. 2, pp. 373–396.Google Scholar
[Ham98a]Gitik, Moti and Shelah, Saharon, Destruction or preservation as you like it, Annals of Pure and Applied Logic, vol. 91 (1998), no. 2–3, pp. 191–229.Google Scholar
[Ham98b]Gitik, Moti and Shelah, Saharon, Small forcing makes any cardinal superdestructible, this Journal, vol. 63 (1998), no. 1, pp. 51–58.Google Scholar
[Ham00]Gitik, Moti and Shelah, Saharon, The lottery preparation, Annals of Pure and Applied Logic, vol. 101 (2000), no. 2–3, pp. 103–146.Google Scholar
[Ham]Gitik, Moti and Shelah, Saharon, Gap forcing, Israel Journal of Mathematics, vol. 125 (2001), pp. 237–252.Google Scholar
[HS98]Hamkins, Joel David and Shelah, Saharon, Superdestructibility: a dual to Laver's indestructibility, this Journal, vol. 63 (1998), no. 2, pp. 549–554, [HmSh:618].Google Scholar
[JMMP80]Jech, T., Magidor, M., Mitchell, W., and Prikry, K., Precipitous ideals, this Journal, vol. 45 (1980), pp. 1–8.Google Scholar
[JW85]Jech, Thomas and Woodin, W. Hugh, Saturation of the closed unbounded filter on the set of regular cardinals, Transactions of the American Mathematical Society, vol. 292 (1985), pp. 345–356.CrossRefGoogle Scholar
[Lav78]Laver, Richard, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385–388.CrossRefGoogle Scholar
[LS67]Lévy, Azriel and Solovay, Robert M., Measurable cardinals and the Continuum Hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234–248.CrossRefGoogle Scholar
[Mag76]Magidor, Menachem, How large is the first strongly compact cardinal?, Annals of Mathematical Logic, vol. 10 (1976), no. 1, pp. 33–57.CrossRefGoogle Scholar
[Mag79]Magidor, Menachem, On the existence of nonregular ultrafilters and the cardinality of ultrapowers, Transactions of the American Mathematical Society, vol. 249 (1979), pp. 97–111.CrossRefGoogle Scholar
[Men74]Menas, Telis K., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974), pp. 327–359.CrossRefGoogle Scholar