Geodesic
On the theory of models for intuitionistic logic
Sbornik. Mathematics, Tome 48 (1984) no. 1, pp. 223-235

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A model approach is developed for intuitionistic and classical theories in the language of the simple theory of types. A weak form of completeness theorem is proved, of which the Henkin completeness theorem is a special case. An ultraproduct of Kripke structures is introduced, and an application of the omitting types theorem to theories with infinite rules for deduction is given. Bibliography: 14 titles. 
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V. I. Stepanov. On the theory of models for intuitionistic logic. Sbornik. Mathematics, Tome 48 (1984) no. 1, pp. 223-235. http://geodesic.mathdoc.fr/item/SM_1984_48_1_a13/