On the complexity of classes of uncountable structures: trees on $\aleph _1$
Volume 253 / 2021
Abstract
We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire space $\omega _1^{\omega_1} $. First, we show that none of these classes have the Baire property (unless they are empty). Moreover, under $V=L$, (a) the class of Aronszajn and Suslin trees is $\Pi^1_1 $-complete, (b) the class of special Aronszajn trees is $\Sigma^1_1 $-complete, and (c) the class of Kurepa trees is $\Pi ^1_2$-complete. We achieve these results by finding nicely definable reductions that map subsets $X$ of ${\omega_1} $ to trees $T_X$ so that $T_X$ is in a given tree class $\mathcal T$ if and only if $X$ is stationary/non-stationary (depending on the class $\mathcal T$). Finally, we present models of CH where these classes have lower projective complexity.
