# On the complexity of classes of uncountable structures: trees on $\aleph_1$

## TL;DR

This paper investigates the descriptive set-theoretic complexity of classes of special uncountable trees, establishing their completeness in the projective hierarchy under certain set-theoretic assumptions.

## Contribution

It provides the first detailed analysis of the projective complexity of classes of Aronszajn, Suslin, and Kurepa trees, including completeness results and definable reductions.

## Key findings

- Aronszajn and Suslin trees are $oldsymbol{ ext{Pi}}_1^1$-complete under $(V=L)$
- Special Aronszajn trees are $oldsymbol{ ext{Sigma}}_1^1$-complete under $(V=L)$
- Kurepa trees are $oldsymbol{ ext{Pi}}_2^1$-complete under $(V=L)$

## 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 will 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.