Geodesic
Index sets of prime model
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 200-210

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We prove that for every nontrivial language $\sigma$, the index set of the class of $d$-decidable prime computable models of $\sigma$ is $\Sigma_3^{0,d}\setminus\Sigma_3^{0,d}$-complete and that the index set of all prime computable models of $\sigma$ is $\Pi_{\omega+2}^0$-complete.
Keywords: prime model, computable model, index sets, model-theoretic constructions, hyperarithmetic hierarchy. 
@article{SEMR_2008_5_a15,
     author = {E. N. Pavlovskii},
     title = {Index sets of prime model},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {200--210},
     publisher = {mathdoc},
     volume = {5},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2008_5_a15/}
}
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E. N. Pavlovskii. Index sets of prime model. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 200-210. http://geodesic.mathdoc.fr/item/SEMR_2008_5_a15/