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Thin Stationary Sets and Disjoint Club Sequences

Sy-David Friedman and John Krueger
Transactions of the American Mathematical Society
Vol. 359, No. 5 (May, 2007), pp. 2407-2420
Stable URL: http://www.jstor.org/stable/20161681
Page Count: 14
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Thin Stationary Sets and Disjoint Club Sequences
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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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