The two-cardinals transfer property

and resurrection of supercompactness

Authors:
Shai Ben-David and Saharon Shelah

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2827-2837

MSC (1991):
Primary 03E35, 03E55, 04A20

DOI:
https://doi.org/10.1090/S0002-9939-96-03327-8

MathSciNet review:
1326996

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the transfer property for singular does not imply (even) the existence of a non-reflecting stationary subset of . The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of ``resurrection of supercompactness''. Our forcing extension destroys the supercompactness of some cardinals; to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extension.

**[BD86]**S. Ben David,*Full reflection of stationary sets*, Abstracts Amer. Math. Soc.**7**(1986), 200.**[BdSh:203]**S. Ben David and S. Shelah,*Souslin trees and successors of singular cardinals*, Ann. Pure Appl. Logic**30**(1985), 207--217. MR**87h:03078****[BM86]**S. Ben David and M. Magidor,*The weak is really weaker than the full*, J. Symbolic Logic**51**(1986), 1029--1033. MR**88a:03117****[BdSh:236]**S. Ben David and S. Shelah,*Non-special Aronszajn trees on*, Israel J. Math.**53**(1986), 93--96. MR**87k:03051****[BD86]**S. Ben David,*A Laver-type indestructability for accessible cardinals*, Logic Colloquium '86 (F. R. Drake and J. K. Truss, eds.), North-Holland, Amsterdam, 1988, pp. 9--19. MR**89e:03093****[E80]**P. Eklof,*Set theoretic methods in homological algebra and abelian groups*, Presses Univ. Montréal, Montréal, 1980. MR**81j:20004****[F83]**W. G. Fleissner,*If all Moore spaces are metrizable, then there is an inner model with a measurable cardinal*, Trans. Amer. Math. Soc.**273**(1982), 365--373. MR**84h:03118****[G76]**J. Gregory,*Higher Souslin trees and the generalized continuum hypothesis*, J. Symbolic Logic**41**(1976), 663--671. MR**58:5208****[J72]**R. B. Jensen,*The fine structure of the constructable hierarchy*, Ann. Math. Logic**4**(1972), 229--308. MR**46:8834****[K78]**K. Kunen,*Saturated Ideals*, J. Symbolic Logic**43**(1978), 65--76. MR**80a:03068****[L78]**R. Laver,*Making the supercompactness of indestructible under -directed closed forcing*, Israel J. Math.**29**(1978), 385--388. MR**57:12226****[M82]**M. Magidor,*Reflecting stationary sets*, J. Symbolic Logic**47**(1982), 755--771. MR**84f:03046****[Mi72]**W. Mitchell,*Aronszajn trees and the independence of the transfer property*, Ann. Math. Logic**5**(1972), 21--46. MR**47:1612****[Sh:44]**S. Shelah,*Infinite abelian groups, Whitehead problem and some constructions*, Israel J. Math.**18**(1974), 243--256. MR**50:9582****[Sh:52]**------,*A compactness theorem for singular cardinals for free algebras. Whitehead problem and transversals*, Israel J. Math.**21**(1975), 329--349. MR**52:10410****[Sh:108]**------,*On the successors of singular cardinals*, Logic Colloquium '78 (M. Boffa, D. Van Dalen, and K. McAloon, eds.), North-Holland, Amsterdam, 1979, pp. 357--380. MR**82d:03079****[Sh:269]**------, ``*Gap*1''*two cardinal principle and omitting type theorems for*, Israel J. Math.**65**(1989), 133--152. MR**91a:03083****[Sh:351]**------,*Reflecting stationary sets and successors of singular cardinals*, Arch. Math. Logic**31**(1991), 25--53. MR**93h:03072**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
03E35,
03E55,
04A20

Retrieve articles in all journals with MSC (1991): 03E35, 03E55, 04A20

Additional Information

**Shai Ben-David**

Affiliation:
Department of Computer Science, Technion-Israel Institute of Technology, Haifa, Israel

Email:
shai@cs.technion.ac.il

**Saharon Shelah**

Affiliation:
Department of Mathematics, The Hebrew University, Jerusalem, Israel

DOI:
https://doi.org/10.1090/S0002-9939-96-03327-8

Received by editor(s):
December 14, 1989

Received by editor(s) in revised form:
March 13, 1995

Communicated by:
Andreas R. Blass

Article copyright:
© Copyright 1996
American Mathematical Society