On the strong equality between supercompactness and strong compactness
Authors:
Arthur W. Apter and Saharon Shelah
Journal:
Trans. Amer. Math. Soc. 349 (1997), 103-128
MSC (1991):
Primary 03E35; Secondary 03E55.
DOI:
https://doi.org/10.1090/S0002-9947-97-01531-6
MathSciNet review:
1333385
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if $V \models$ ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension $V[{G}] \models$ ZFC + GCH in which, (a) (preservation) for $\kappa \le \lambda$ regular, if $V \models {}$ “$\kappa$ is $\lambda$ supercompact”, then $V[G] \models {}$ “$\kappa$ is $\lambda$ supercompact” and so that, (b) (equivalence) for $\kappa \le \lambda$ regular, $V[{G}] \models {}$ “$\kappa$ is $\lambda$ strongly compact” iff $V[{G}] \models {}$ “$\kappa$ is $\lambda$ supercompact”, except possibly if $\kappa$ is a measurable limit of cardinals which are $\lambda$ supercompact.
- Arthur W. Apter, On the least strongly compact cardinal, Israel J. Math. 35 (1980), no. 3, 225–233. MR 576474, DOI https://doi.org/10.1007/BF02761194
- A. Apter, S. Shelah, “Menas’ Result is Best Possible”, Trans. Amer. Math. Soc. (to appear).
- James E. Baumgartner, Iterated forcing, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1–59. MR 823775, DOI https://doi.org/10.1017/CBO9780511758867.002
- Set theory, Handbook of mathematical logic, Part B, North-Holland, Amsterdam, 1977, pp. 317–522. Studies in Logic and Foundations of Math., Vol. 90. With contributions by J. R. Shoenfield, Thomas J. Jech, Kenneth Kunen, John P. Burgess, Keith J. Devlin, Mary Ellen Rudin and I. Juhász. MR 0540758
- 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, DOI https://doi.org/10.1090/S0002-9947-1992-1041044-9
- J. Cummings, H. Woodin, Generalised Prikry Forcings, circulated manuscript of a forthcoming book.
- Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. MR 506523
- 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
- Y. Kimchi, M. Magidor, “The Independence between the Concepts of Compactness and Supercompactness”, circulated manuscript.
- Menachem Magidor, How large is the first strongly compact cardinal? or A study on identity crises, Ann. Math. Logic 10 (1976), no. 1, 33–57. MR 429566, DOI https://doi.org/10.1016/0003-4843%2876%2990024-3
- M. Magidor, On the role of supercompact and extendible cardinals in logic, Israel J. Math. 10 (1971), 147–157. MR 295904, DOI https://doi.org/10.1007/BF02771565
- M. Magidor, There are many normal ultrafiltres corresponding to a supercompact cardinal, Israel J. Math. 9 (1971), 186–192. MR 347607, DOI https://doi.org/10.1007/BF02771583
- M. Magidor, unpublished; personal communication.
- Telis K. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974/75), 327–359. MR 357121, DOI https://doi.org/10.1016/0003-4843%2875%2990009-1
- Alan H. Mekler and Saharon Shelah, When $\kappa $-free implies strongly $\kappa $-free, Abelian group theory (Oberwolfach, 1985) Gordon and Breach, New York, 1987, pp. 137–148. MR 1011309
- 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, DOI https://doi.org/10.1016/0003-4843%2878%2990031-1
Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 03E35, 03E55.
Retrieve articles in all journals with MSC (1991): 03E35, 03E55.
Additional Information
Arthur W. Apter
Affiliation:
Department of Mathematics, Baruch College of CUNY, New York, New York 10010
MR Author ID:
26680
Email:
awabb@cunyvm.cuny.edu
Saharon Shelah
Affiliation:
Department of Mathematics, The Hebrew University, Jerusalem, Israel;
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08904
MR Author ID:
160185
ORCID:
0000-0003-0462-3152
Email:
shelah@sunrise.huji.ac.il, shelah@math.rutgers.edu
Keywords:
Strongly compact cardinal,
supercompact cardinal.
Received by editor(s):
May 2, 1994
Received by editor(s) in revised form:
December 30, 1994
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 The Hebrew University and Tel Aviv University for the hospitality shown him during his sabbatical in Israel. The second author wishes to thank the Basic Research Fund of the Israeli Academy of Sciences for partially supporting this research, which is Publication 495 of the second author.
Article copyright:
© Copyright 1997
American Mathematical Society