Weakly compact cardinals and nonspecial Aronszajn trees

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.