Some consequences of reflection on the approachability ideal
HTML articles powered by AMS MathViewer
- by Assaf Sharon and Matteo Viale PDF
- Trans. Amer. Math. Soc. 362 (2010), 4201-4212 Request permission
Abstract:
We study the approachability ideal $\mathcal {I}[\kappa ^+]$ in the context of large cardinals and properties of the regular cardinals below a singular $\kappa$. As a guiding example consider the approachability ideal $\mathcal {I}[\aleph _{\omega +1}]$ assuming that $\aleph _\omega$ is a strong limit. In this case we obtain that club many points in $\aleph _{\omega +1}$ of cofinality $\aleph _n$ for some $n>1$ are approachable assuming the joint reflection of countable families of stationary subsets of $\aleph _n$. This reflection principle holds under $\mathsf {MM}$ for all $n>1$ and for each $n>1$ is equiconsistent with $\aleph _n$ being weakly compact in $L$. This characterizes the structure of the approachability ideal $\mathcal {I}[\aleph _{\omega +1}]$ in models of $\mathsf {MM}$. We also apply our result to show that the Chang conjecture $(\kappa ^+,\kappa )\twoheadrightarrow (\aleph _2,\aleph _1)$ fails in models of $\mathsf {MM}$ for all singular cardinals $\kappa$.References
- U. Abraham and M. Magidor, Cardinal arithmetic, Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), North Holland, to appear.
- James Cummings, Collapsing successors of singulars, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2703–2709. MR 1416080, DOI 10.1090/S0002-9939-97-03995-6
- T. Eisworth, Successors of singular cardinals, Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), North Holland, to appear.
- M. Foreman, Ideals and Generic Elementary Embeddings, Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), North Holland, to appear.
- Matthew Foreman and Menachem Magidor, A very weak square principle, J. Symbolic Logic 62 (1997), no. 1, 175–196. MR 1450520, DOI 10.2307/2275738
- 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, DOI 10.2307/1971415
- Moti Gitik and Assaf Sharon, On SCH and the approachability property, Proc. Amer. Math. Soc. 136 (2008), no. 1, 311–320. MR 2350418, DOI 10.1090/S0002-9939-07-08716-3
- Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
- Menachem Kojman, Exact upper bounds and their uses in set theory, Ann. Pure Appl. Logic 92 (1998), no. 3, 267–282. MR 1640912, DOI 10.1016/S0168-0072(98)00011-6
- Jean-Pierre Levinski, Menachem Magidor, and Saharon Shelah, Chang’s conjecture for $\aleph _\omega$, Israel J. Math. 69 (1990), no. 2, 161–172. MR 1045371, DOI 10.1007/BF02937302
- Menachem Magidor, Reflecting stationary sets, J. Symbolic Logic 47 (1982), no. 4, 755–771 (1983). MR 683153, DOI 10.2307/2273097
- Ernest Schimmerling, Coherent sequences and threads, Adv. Math. 216 (2007), no. 1, 89–117. MR 2353251, DOI 10.1016/j.aim.2007.05.005
- Stevo Todorcevic, Walks on ordinals and their characteristics, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007. MR 2355670, DOI 10.1007/978-3-7643-8529-3
- M. Viale, Application of the proper forcing axiom to cardinal arithmetic, Ph.D. thesis, Université Paris 7-Denis Diderot, 2006.
- Matteo Viale, A family of covering properties, Math. Res. Lett. 15 (2008), no. 2, 221–238. MR 2385636, DOI 10.4310/MRL.2008.v15.n2.a2
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
- MR Author ID: 719238
- Email: matteo.viale@unito.it
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4201-4212
- MSC (2000): Primary 03E04, 03E55; Secondary 03E65
- DOI: https://doi.org/10.1090/S0002-9947-10-04976-7
- MathSciNet review: 2608402