On \Sigma^1_1-complete Equivalence Relations on the Generalized Baire Space

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V = L, then many of them are \Sigma^1_1-complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in ZFC whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is \Sigma^1_1-complete (it is, if V = L, but can be forced not to be).Comment: 22 page

On the complexity of classes of uncountable structures: trees on ℵ1\aleph_1

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.Comment: 16 page

Finding bases of uncountable free abelian groups is usually difficult

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption $V=L$, that there is a first-order definable free abelian group with no first-order definable basis.Comment: 26 page