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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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: 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$.

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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

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.