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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Thin stationary sets and disjoint club sequences


Authors: Sy-David Friedman and John Krueger
Journal: Trans. Amer. Math. Soc. 359 (2007), 2407-2420
MSC (2000): Primary 03E35, 03E40
Published electronically: December 5, 2006
MathSciNet review: 2276627
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Abstract: We describe two opposing combinatorial properties related to adding clubs to $ \omega_2$: the existence of a thin stationary subset of $ P_{\omega_1}(\omega_2)$ and the existence of a disjoint club sequence on $ \omega_2$. A special Aronszajn tree on $ \omega_2$ implies there exists a thin stationary set. If there exists a disjoint club sequence, then there is no thin stationary set, and moreover there is a fat stationary subset of $ \omega_2$ which cannot acquire a club subset by any forcing poset which preserves $ \omega_1$ and $ \omega_2$. We prove that the existence of a disjoint club sequence follows from Martin's Maximum and is equiconsistent with a Mahlo cardinal.


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Additional Information

Sy-David Friedman
Affiliation: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Waehringer Strasse 25, A-1090 Wien, Austria
Email: sdf@logic.univie.ac.at

John Krueger
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: jkrueger@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04163-8
PII: S 0002-9947(06)04163-8
Received by editor(s): June 28, 2005
Published electronically: December 5, 2006
Additional Notes: The authors were supported by FWF project number P16790-N04.
Article copyright: © Copyright 2006 American Mathematical Society