Geodesic
Complete Theories with Finitely Many Countable Models.~I
Algebra i logika, Tome 43 (2004) no. 1, pp. 110-124.

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A syntactic characterization is furnished for the class of elementary complete theories with finitely many countable models, which is the analog of a known theorem by Ryll-Nardzewski on countably categorical theories, and is based on classifying the theories by Rudin–Keisler quasiorders and distribution functions of a number of models limit over types.
Keywords: elementary complete theory, countable model, Rudin–Keisler quasiorder. 
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S. V. Sudoplatov. Complete Theories with Finitely Many Countable Models.~I. Algebra i logika, Tome 43 (2004) no. 1, pp. 110-124. http://geodesic.mathdoc.fr/item/AL_2004_43_1_a4/

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