  We classify the computability-theoretic complexity of two index sets of
classes of first-order theories: We show that the property of being an
$\aleph_0$-categorical theory is $\Pi^0_3$-complete; and the property of being
an Ehrenfeucht theory $\Pi^1_1$-complete. We also show that the property of
having continuum many models is $\Sigma^1_1$-hard. Finally, as a corollary, we
note that the properties of having only decidable models, and of having only
computable models, are both $\Pi^1_1$-complete.
