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Weakly compact cardinals and nonspecial Aronszajn trees


Authors: Saharon Shelah and Lee Stanley
Journal: Proc. Amer. Math. Soc. 104 (1988), 887-897
MSC: Primary 03E05; Secondary 03E45, 03E55, 04A20
DOI: https://doi.org/10.1090/S0002-9939-1988-0964870-5
MathSciNet review: 964870
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Abstract: Lemma 1. If $ \lambda $ is a cardinal with cf $ \lambda > \omega $, then $ {\square _\lambda }$ implies that there is a $ {\lambda ^ + }$-Aronszajn tree with an $ \omega $-ascent path, i.e. a sequence $ \left( {{{\bar x}^\alpha }:\alpha < {\lambda ^ + }} \right)$ with each $ {\bar x^\alpha } = \left( {x_n^\alpha :n < \omega } \right)$ a one-to-one sequence from $ {T_\alpha }$, such that for all $ \alpha < \beta < {\lambda ^ + },x_n^\alpha $ precedes $ x_n^\beta $ in the tree order for sufficiently large $ n$.

Lemma 2. If $ \lambda $ is a cardinal with $ \operatorname{cf} \lambda = \omega < \lambda $, then $ {\square _\lambda }$ implies that there is a $ {\lambda ^ + }$-Aronszajn tree with an $ {\omega _1}$-ascent path (replace $ \omega $ by $ {\omega _1}$, above).

Lemma 3. If $ \lambda $ is an uncountable cardinal, $ \kappa $ is regular, $ \kappa < \lambda ,\operatorname{cf} \lambda \ne \kappa ,T$ is a $ {\lambda ^ + }$-Aronszajn tree, and $ \left( {x_i^\alpha :i < \kappa } \right)$ is a one-to-one sequence from $ {T_{\zeta \left( \alpha \right)}}$ with the property of ascent paths, where $ \zeta :{\lambda ^ + } \to {\lambda ^ + }$ is a monotone increasing function of $ \alpha $, then $ T$ is nonspecial.

Theorem 4. If $ \lambda $ is uncountable, then $ {\square _\lambda }$ implies that there is a nonspecial $ {\lambda ^ + }$-Aronszajn tree.

Theorem 5. If $ \lambda $ is an uncountable cardinal, $ \kappa = {\lambda ^ + }$, and $ \kappa $ is not $ {(weakly\;compact)^L}$, then there is a nonspecial $ \kappa $-Aronszajn tree.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0964870-5
Article copyright: © Copyright 1988 American Mathematical Society

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