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We classify the computability-theoretic complexity of two index sets of classes of first-order theories: We show that the property of being an _0-categorical theory is Π^0_3-complete; and the property of being an Ehrenfeucht theory Π^1_1-complete. We also show that the property of having continuum many models is Σ^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 Π^1_1-complete.arXiv:math/0610776v1 fatcat:l4ud2baig5d75j62hq324ggnpy