Abstract.
In this paper, we first prove several general theorems about strongness, supercompactness, and indestructibility, along the way giving some new applications of Hamkins’ lottery preparation forcing to indestructibility. We then show that it is consistent, relative to the existence of cardinals κ<λ so that κ is λ supercompact and λ is inaccessible, for the least strongly compact cardinal κ to be the least strong cardinal and to have its strongness, but not its strong compactness, indestructible under κ-strategically closed forcing.
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Apter, A.: A note on strong compactness and resurrectibility. Fundamenta Mathematicae 165, 285–290 (2000)
Apter, A.: Aspects of strong compactness, measurability, and indestructibility. Arch. Math. Logic 41, 705–719 (2002)
Apter, A.: Indestructibility and strong compactness. in preparation.
Apter, A.: Laver indestructibility and the class of compact cardinals. J. Symbolic Logic 63, 149–157 (1998)
Apter, A.: Strong cardinals can be fully Laver indestructible. Math. Logic Quarterly 48, 499–507 (2002)
Apter, A.: Strong compactness, measurability, and the class of supercompact cardinals. Fundamenta Mathematicae 167, 65–78 (2001)
Apter, A., Cummings, J.: Identity crises and strong compactness II: Strong cardinals. Arch. Math. Logic 40, 25–38 (2001)
Apter, A., Gitik, M.: The least measurable can be strongly compact and indestructible. J. Symbolic Logic 63, 1404–1412 (1998)
Apter, A., Hamkins, J.D.: Exactly controlling the non-supercompact strongly compact cardinals. To appear in the J. Symbolic Logic.
Apter, A., Hamkins, J.D.: Universal indestructibility. Kobe J. Math. 16, 119–130 (1999)
Burgess, J.: Forcing. In: J. Barwise, editor, Handbook of Mathematical Logic, North-Holland, Amsterdam, 403–452 (1977)
Cummings, J.: A model in which GCH holds at successors but fails at limits. Trans. Amer. Math. Soc. 329, 1–39 (1992)
Felgner, U.: Comparison of the axioms of local and universal choice. Fundamenta Mathematicae 71, 62–73 (1971)
Foreman, M.: More saturated ideals. In: Cabal Seminar 79–81, Lecture Notes in Mathematics 1019, Springer-Verlag, Berlin and New York, 1–27 (1983)
Gitik, M.: All uncountable cardinals can be singular. Israel J. Math. 35, 61–88 (1980)
Gitik, M., Shelah, S.: On certain indestructibility of strong cardinals and a question of Hajnal. Arch. Math. Logic 28, 35–42 (1989)
Hamkins, J.D.: A class of strong diamond principles. Submitted for publication to Ann. Pure and Applied Logic.
Hamkins, J.D.: Destruction or preservation as you like it. Ann. Pure and Applied Logic 91, 191–229 (1998)
Hamkins, J.D.: Gap forcing. Israel J. Math. 125, 237–252 (2001)
Hamkins, J.D.: Gap forcing: Generalizing the Lévy-Solovay Theorem. Bull. Symbolic Logic 5, 264–272 (1999)
Hamkins, J.D.: The lottery preparation. Ann. Pure and Applied Logic 101, 103–146 (2000)
Kanamori, A.: The higher infinite. Springer–Verlag, Berlin and New York (1994)
Kimchi, Y., Magidor, M.: The independence between the concepts of compactness and supercompactness. Circulated manuscript.
Laver, R.: Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel J. Math. 29, 385–388 (1978)
Martin, D.A., Steel, J.: A proof of projective determinacy. J. Amer. Math. Soc. 2, 71–125 (1989)
Solovay, R., Reinhardt, W., Kanamori, A.: Strong axioms of infinity and elementary embeddings. Ann. Math. Logic 13, 73–116 (1978)
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Mathematics Subject Classification (2000): 03E35, 03E55
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Apter, A. Some remarks on indestructibility and Hamkins’ lottery preparation. Arch. Math. Logic 42, 717–735 (2003). https://doi.org/10.1007/s00153-003-0181-3
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DOI: https://doi.org/10.1007/s00153-003-0181-3