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Complexity of Index Sets for Several Classes of Models
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 8 (2008) no. 1, pp. 71-76 Cet article a éte moissonné depuis la source Math-Net.Ru

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When we study questions about computable characterization existence for different classes of models the approach suggested by Goncharov and Knight [1] is effective. It consists of obtaining precise estimations of index sets of such classes in corresponding hierarchy. For the universal numeration of computable models in non-trivial computable language there were found precise estimations of the following index sets of computable models classes: models with Ehrenfeucht theory ($\Pi^1_1$), models with a theory admitting infinitely many countable models ($\Sigma^1_1$). 
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E. N. Pavlovsky. Complexity of Index Sets for Several Classes of Models. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 8 (2008) no. 1, pp. 71-76. http://geodesic.mathdoc.fr/item/VNGU_2008_8_1_a6/

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