Menas' Result is Best Possible
Authors: Arthur W. Apter and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 349 (1997), 2007-2034
MSC (1991): Primary 03E55; Secondary 03E35
MathSciNet review: 1370634
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Abstract: Generalizing some earlier techniques due to the second author, we show that Menas' theorem which states that the least cardinal which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author.
- [AS] A. Apter, S. Shelah, On the Strong Equality between Supercompactness and Strong Compactness, Transactions AMS 349 (1997), 103-128.
- [Ba] A. R. D. Mathias (ed.), Surveys in set theory, London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983. MR 823774
- [Bu] Handbook of mathematical logic, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Edited by Jon Barwise; With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra; Studies in Logic and the Foundations of Mathematics, Vol. 90. MR 0457132
- [C] James Cummings, A model in which GCH holds at successors but fails at limits, Trans. Amer. Math. Soc. 329 (1992), no. 1, 1–39. MR 1041044, https://doi.org/10.1090/S0002-9947-1992-1041044-9
- [CW] J. Cummings, H. Woodin, Generalised Prikry Forcings, circulated manuscript of a forthcoming book.
- [J] Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. MR 506523
- [Ka] Akihiro Kanamori, The higher infinite, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994. Large cardinals in set theory from their beginnings. MR 1321144
- [KaM] A. Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Math., vol. 669, Springer, Berlin, 1978, pp. 99–275. MR 520190
- [KiM] Y. Kimchi, M. Magidor, The Independence between the Concepts of Compactness and Supercompactness, circulated manuscript.
- [L] Richard Laver, Making the supercompactness of 𝜅 indestructible under 𝜅-directed closed forcing, Israel J. Math. 29 (1978), no. 4, 385–388. MR 0472529, https://doi.org/10.1007/BF02761175
- [LS] A. Lévy and R. M. Solovay, Measurable cardinals and the continuum hypothesis, Israel J. Math. 5 (1967), 234–248. MR 0224458, https://doi.org/10.1007/BF02771612
- [Ma1] Menachem Magidor, Changing cofinality of cardinals, Fund. Math. 99 (1978), no. 1, 61–71. MR 0465868
- [Ma2] M. Magidor, On the role of supercompact and extendible cardinals in logic, Israel J. Math. 10 (1971), 147–157. MR 0295904, https://doi.org/10.1007/BF02771565
- [Me] Telis K. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974/75), 327–359. MR 0357121, https://doi.org/10.1016/0003-4843(75)90009-1
- [MS] Alan H. Mekler and Saharon Shelah, When 𝜅-free implies strongly 𝜅-free, Abelian group theory (Oberwolfach, 1985) Gordon and Breach, New York, 1987, pp. 137–148. MR 1011309
- [SRK] Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), no. 1, 73–116. MR 482431, https://doi.org/10.1016/0003-4843(78)90031-1
- A. Apter, S. Shelah, On the Strong Equality between Supercompactness and Strong Compactness, Transactions AMS 349 (1997), 103-128.
- J. Baumgartner, Iterated Forcing, Surveys in Set Theory (A. Mathias, ed.), Cambridge University Press, Cambridge, England, 1983, pp. 1-59. MR 86m:03005
- J. Burgess, Forcing, Handbook of Mathematical Logic (J. Barwise, ed.), North-Holland, Amsterdam, 1977, pp. 403-452. MR 56:15351
- J. Cummings, A Model in which GCH Holds at Successors but Fails at Limits, Transactions AMS 329 (1992), 1-39. MR 92h:03076
- J. Cummings, H. Woodin, Generalised Prikry Forcings, circulated manuscript of a forthcoming book.
- T. Jech, Set Theory, Academic Press, New York, 1978. MR 80a:03062
- A. Kanamori, The Higher Infinite, Springer-Verlag, New York and Berlin, 1994. MR 96k:03125
- A. Kanamori, M. Magidor, The Evolution of Large Cardinal Axioms in Set Theory, Lecture Notes in Mathematics, vol. 669, Springer-Verlag, Berlin, 1978, pp. 99-275. MR 80b:03083
- Y. Kimchi, M. Magidor, The Independence between the Concepts of Compactness and Supercompactness, circulated manuscript.
- R. Laver, Making the Supercompactness of Indestructible under -Directed Closed Forcing, Israel J. Math. 29 (1978), 385-388. MR 57:12226
- A. Lévy, R. Solovay, Measurable Cardinals and the Continuum Hypothesis, Israel J. Math. 5 (1967), 234-248. MR 37:57
- M. Magidor, Changing Cofinality of Cardinals, Fundamenta Mathematicae 99 (1978), 61-71. MR 57:5754
- -, On the Role of Supercompact and Extendible Cardinals in Logic, Israel J. Math. 10 (1971), 147-157. MR 45:4966
- T. Menas, On Strong Compactness and Supercompactness, Annals Math. Logic 7 (1975), 327-359. MR 50:9589
- A. Mekler, S. Shelah, Does -Free Imply Strongly -Free?, Proceedings of the Third Conference on Abelian Group Theory, Gordon and Breach, Salzburg, 1987, pp. 137-148. MR 90f:20082
- R. Solovay, W. Reinhardt, A. Kanamori, Strong Axioms of Infinity and Elementary Embeddings, Annals Math. Logic 13 (1978), 73-116. MR 80h:03072
Arthur W. Apter
Affiliation: Department of Mathematics, Baruch College of CUNY, New York, New York 10010
Affiliation: Department of Mathematics, The Hebrew University, Jerusalem, Israel; Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08904
Email: email@example.com, firstname.lastname@example.org
Received by editor(s): March 3, 1995
Received by editor(s) in revised form: December 13, 1995
Additional Notes: The research of the first author was partially supported by PSC-CUNY Grant 662341 and a salary grant from Tel Aviv University. In addition, the first author wishes to thank the Mathematics Departments of Hebrew University and Tel Aviv University for the hospitality shown him during his sabbatical in Israel.
Publication 496. The second author wishes to thank the Basic Research Fund of the Israeli Academy of Sciences for partially supporting this research.
Article copyright: © Copyright 1997 American Mathematical Society