Weakly compact cardinals and nonspecial Aronszajn trees

Abstract

Lemma 1. If λ \lambda is a cardinal with cf λ > ω \lambda > \omega , then ◻ λ {\square _\lambda } implies that there is a λ + {\lambda ^ + } -Aronszajn tree with an ω \omega -ascent path, i.e. a sequence ( x ¯ α : α > λ + ) \left ( {{{\bar x}^\alpha }:\alpha > {\lambda ^ + }} \right ) with each x ¯ α = ( x n α : n > ω ) {\bar x^\alpha } = \left ( {x_n^\alpha :n > \omega } \right ) a one-to-one sequence from T α {T_\alpha } , such that for all α > β > λ + , x n α \alpha > \beta > {\lambda ^ + },x_n^\alpha precedes x n β x_n^\beta in the tree order for sufficiently large n n . Lemma 2. If λ \lambda is a cardinal with cf ⁡ λ = ω > λ \operatorname {cf} \lambda = \omega > \lambda , then ◻ λ {\square _\lambda } implies that there is a λ + {\lambda ^ + } -Aronszajn tree with an ω 1 {\omega _1} -ascent path (replace ω \omega by ω 1 {\omega _1} , above). Lemma 3. If λ \lambda is an uncountable cardinal, κ \kappa is regular, κ > λ , cf ⁡ λ ≠ κ , T \kappa > \lambda ,\operatorname {cf} \lambda \ne \kappa ,T is a λ + {\lambda ^ + } -Aronszajn tree, and ( x i α : i > κ ) \left ( {x_i^\alpha :i > \kappa } \right ) is a one-to-one sequence from T ζ ( α ) {T_{\zeta \left ( \alpha \right )}} with the property of ascent paths, where ζ : λ + → λ + \zeta :{\lambda ^ + } \to {\lambda ^ + } is a monotone increasing function of α \alpha , then T T is nonspecial. Theorem 4. If λ \lambda is uncountable, then ◻ λ {\square _\lambda } implies that there is a nonspecial λ + {\lambda ^ + } -Aronszajn tree. Theorem 5. If λ \lambda is an uncountable cardinal, κ = λ + \kappa = {\lambda ^ + } , and κ \kappa is not ( w e a k l y c o m p a c t ) L {(weakly\;compact)^L} , then there is a nonspecial κ \kappa -Aronszajn tree.