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

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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 ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension ZFC + GCH in which, (a) (preservation) for regular, if is supercompact'', then is supercompact'' and so that, (b) (equivalence) for regular, is strongly compact'' iff is supercompact'', except possibly if is a measurable limit of cardinals which are supercompact.

**[A]**A. Apter,*``On the Least Strongly Compact Cardinal"*, Israel J. Math.**35**, 1980, 225-233. MR**81h:03099****[AS]**A. Apter, S. Shelah,*``Menas' Result is Best Possible''*, Trans. Amer. Math. Soc. (to appear).**[Ba]**J. Baumgartner,*``Iterated Forcing"*, in: A. Mathias, ed., Surveys in Set Theory, Cambridge University Press, Cambridge, England, 1983, 1-59. MR**87c:03099****[Bu]**J. Burgess,*``Forcing"*, in: J. Barwise, ed., Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, 403-452. MR**58:27475****[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****[KaM]**A. Kanamori, M. Magidor,*``The Evolution of Large Cardinal Axioms in Set Theory"*, in: Lecture Notes in Mathematics**669**, Springer-Verlag, Berlin, 1978, 99-275. MR**80b:03083****[KiM]**Y. Kimchi, M. Magidor,*``The Independence between the Concepts of Compactness and Supercompactness''*, circulated manuscript.**[Ma1]**M. Magidor,*``How Large is the First Strongly Compact Cardinal?"*, Annals Math. Logic**10**, 1976, 33-57. MR**55:2578****[Ma2]**M. Magidor,*``On the Role of Supercompact and Extendible Cardinals in Logic"*, Israel J. Math.**10**, 1971, 147-157. MR**45:4966****[Ma3]**M. Magidor,*``There are Many Normal Ultrafilters Corresponding to a Supercompact Cardinal"*, Israel J. Math.**9**, 1971, 186-192. MR**50:110****[Ma4]**M. Magidor, unpublished; personal communication.**[Me]**T. Menas,*``On Strong Compactness and Supercompactness"*, Annals Math. Logic**7**, 1975, 327-359. MR**50:9589****[MS]**A. Mekler, S. Shelah,*``When -Free Implies Strongly -Free"*, in: Proceedings of the Third Conference on Abelian Group Theory, Gordon and Breach, Salzburg, 1987, 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**

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Additional Information

**Arthur W. Apter**

Affiliation:
Department of Mathematics, Baruch College of CUNY, New York, New York 10010

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

Email:
shelah@sunrise.huji.ac.il, shelah@math.rutgers.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01531-6

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