  We answer two questions posed by Castro and Cucker, giving the exact
complexities of two decision problems about cardinalities of omega-languages of
Turing machines. Firstly, it is $D_2(\Sigma_1^1)$-complete to determine whether
the omega-language of a given Turing machine is countably infinite, where
$D_2(\Sigma_1^1)$ is the class of 2-differences of $\Sigma_1^1$-sets. Secondly,
it is $\Sigma_1^1$-complete to determine whether the omega-language of a given
Turing machine is uncountable.
