The complexity of the index sets of $\aleph_0$-categorical theories and   of Ehrenfeucht theories
  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. 
Steffen Lempp and Theodore A. Slaman