Some consequences of reflection on the approachability ideal

Authors:
Assaf Sharon and Matteo Viale

Journal:
Trans. Amer. Math. Soc. **362** (2010), 4201-4212

MSC (2000):
Primary 03E04, 03E55; Secondary 03E65

Published electronically:
March 8, 2010

MathSciNet review:
2608402

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the approachability ideal in the context of large cardinals and properties of the regular cardinals below a singular . As a guiding example consider the approachability ideal assuming that is a strong limit. In this case we obtain that club many points in of cofinality for some are approachable assuming the joint reflection of countable families of stationary subsets of . This reflection principle holds under for all and for each is equiconsistent with being weakly compact in . This characterizes the structure of the approachability ideal in models of . We also apply our result to show that the Chang conjecture fails in models of for all singular cardinals .

**1.**U. Abraham and M. Magidor,*Cardinal arithmetic*, Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), North Holland, to appear.**2.**James Cummings,*Collapsing successors of singulars*, Proc. Amer. Math. Soc.**125**(1997), no. 9, 2703–2709. MR**1416080**, 10.1090/S0002-9939-97-03995-6**3.**T. Eisworth,*Successors of singular cardinals*, Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), North Holland, to appear.**4.**M. Foreman,*Ideals and Generic Elementary Embeddings*, Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), North Holland, to appear.**5.**Matthew Foreman and Menachem Magidor,*A very weak square principle*, J. Symbolic Logic**62**(1997), no. 1, 175–196. MR**1450520**, 10.2307/2275738**6.**M. Foreman, M. Magidor, and S. Shelah,*Martin’s maximum, saturated ideals, and nonregular ultrafilters. I*, Ann. of Math. (2)**127**(1988), no. 1, 1–47. MR**924672**, 10.2307/1971415**7.**Moti Gitik and Assaf Sharon,*On SCH and the approachability property*, Proc. Amer. Math. Soc.**136**(2008), no. 1, 311–320 (electronic). MR**2350418**, 10.1090/S0002-9939-07-08716-3**8.**Thomas Jech,*Set theory*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR**1940513****9.**Menachem Kojman,*Exact upper bounds and their uses in set theory*, Ann. Pure Appl. Logic**92**(1998), no. 3, 267–282. MR**1640912**, 10.1016/S0168-0072(98)00011-6**10.**Jean-Pierre Levinski, Menachem Magidor, and Saharon Shelah,*Chang’s conjecture for ℵ_{𝜔}*, Israel J. Math.**69**(1990), no. 2, 161–172. MR**1045371**, 10.1007/BF02937302**11.**Menachem Magidor,*Reflecting stationary sets*, J. Symbolic Logic**47**(1982), no. 4, 755–771 (1983). MR**683153**, 10.2307/2273097**12.**Ernest Schimmerling,*Coherent sequences and threads*, Adv. Math.**216**(2007), no. 1, 89–117. MR**2353251**, 10.1016/j.aim.2007.05.005**13.**Stevo Todorcevic,*Walks on ordinals and their characteristics*, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007. MR**2355670****14.**M. Viale,*Application of the proper forcing axiom to cardinal arithmetic*, Ph.D. thesis, Université Paris 7-Denis Diderot, 2006.**15.**Matteo Viale,*A family of covering properties*, Math. Res. Lett.**15**(2008), no. 2, 221–238. MR**2385636**, 10.4310/MRL.2008.v15.n2.a2

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Additional Information

**Assaf Sharon**

Affiliation:
Tarad 11, Apt. 10, 52503 Ramat Gan, Israel

**Matteo Viale**

Affiliation:
Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy

Email:
matteo.viale@unito.it

DOI:
http://dx.doi.org/10.1090/S0002-9947-10-04976-7

Keywords:
Set theory,
singular cardinal combinatorics,
large cardinals

Received by editor(s):
April 16, 2008

Published electronically:
March 8, 2010

Additional Notes:
The second author acknowledges support of the Austrian Science Fund FWF project P19375-N18 for this research. The second author also thanks Boban Veličković for several useful hints and comments on previous drafts. In particular the results in subsection 2.4 are due to him.

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.