Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing some earlier techniques due to the second author, we show that Menas' theorem which states that the least cardinal $\kappa $ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not $2^{\kappa }$ 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these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author.

References [Enhancements On Off] (What's this?)

  • [AS] A. Apter, S. Shelah, On the Strong Equality between Supercompactness and Strong Compactness, Transactions AMS 349 (1997), 103-128.
  • [Ba] J. Baumgartner, Iterated Forcing, Surveys in Set Theory (A. Mathias, ed.), Cambridge University Press, Cambridge, England, 1983, pp. 1-59. MR 86m:03005
  • [Bu] J. Burgess, Forcing, Handbook of Mathematical Logic (J. Barwise, ed.), North-Holland, Amsterdam, 1977, pp. 403-452. MR 56:15351
  • [C] J. Cummings, A Model in which GCH Holds at Successors but Fails at Limits, Transactions AMS 329 (1992), 1-39. MR 92h:03076
  • [CW] J. Cummings, H. Woodin, Generalised Prikry Forcings, circulated manuscript of a forthcoming book.
  • [J] T. Jech, Set Theory, Academic Press, New York, 1978. MR 80a:03062
  • [Ka] A. Kanamori, The Higher Infinite, Springer-Verlag, New York and Berlin, 1994. MR 96k:03125
  • [KaM] 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
  • [KiM] Y. Kimchi, M. Magidor, The Independence between the Concepts of Compactness and Supercompactness, circulated manuscript.
  • [L] R. Laver, Making the Supercompactness of $\k $ Indestructible under $\k $-Directed Closed Forcing, Israel J. Math. 29 (1978), 385-388. MR 57:12226
  • [LS] A. Lévy, R. Solovay, Measurable Cardinals and the Continuum Hypothesis, Israel J. Math. 5 (1967), 234-248. MR 37:57
  • [Ma1] M. Magidor, Changing Cofinality of Cardinals, Fundamenta Mathematicae 99 (1978), 61-71. MR 57:5754
  • [Ma2] -, On the Role of Supercompact and Extendible Cardinals in Logic, Israel J. Math. 10 (1971), 147-157. MR 45:4966
  • [Me] T. Menas, On Strong Compactness and Supercompactness, Annals Math. Logic 7 (1975), 327-359. MR 50:9589
  • [MS] A. Mekler, S. Shelah, Does $\k $-Free Imply Strongly $\k $-Free?, Proceedings of the Third Conference on Abelian Group Theory, Gordon and Breach, Salzburg, 1987, pp. 137-148. MR 90f:20082
  • [SRK] R. Solovay, W. Reinhardt, A. Kanamori, Strong Axioms of Infinity and Elementary Embeddings, Annals Math. Logic 13 (1978), 73-116. MR 80h:03072

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 03E55, 03E35

Retrieve articles in all journals with MSC (1991): 03E55, 03E35

Additional Information

Arthur W. Apter
Affiliation: Department of Mathematics, Baruch College of CUNY, New York, New York 10010

Saharon Shelah
Affiliation: Department of Mathematics, The Hebrew University, Jerusalem, Israel; Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08904

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

American Mathematical Society