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Semantic universality of theories over a superlist. (English. Russian original) Zbl 0780.03013
Algebra Logic 31, No. 1, 30-48 (1992); translation from Algebra Logika 31, No. 1, 47-73 (1992).
Summary: We study the expressive power of formulas of the first-order predicate logic over a list PL of model-theoretic properties, which is more extensive than MTL figuring in two earlier papers [the author, Dokl. Akad. Nauk SSSR 308, No. 4, 788-791 (1989; Zbl 0703.03014); “Immersing recursively enumerable theories in finitely axiomatizable ones” (Russian), Tr. Inst. Mat. (1991)]. Characterization of generalized Lindenbaum L-algebras is obtained for semantically universal finitely axiomatizable model classes under inclusion $$\text{L}\subseteq\text{PL}$$. As an immediate consequence, the existence of a recursive isomorphism (preserving properties from the list PL) between Lindenbaum algebras of the predicate calculus with two diverse finite signatures is proved. This gives a positive answer to question 1 posed by the author in Algebra Logika 30, No. 4, 414-431 (1991; Zbl 0777.03010).
MSC:
 03C57 Computable structure theory, computable model theory
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References:
 [1] M. G. Peretyat’kin, ”Similarity of properties of recursively enumerable and finitely axiomatizable theories,” Dokl. Akad. Nauk SSSR,308, No. 4, 788–791. [2] M. G. Peretyat’kin, ”Immersing recursively enumerable theories in finitely axiomatizable ones,” Tr. Inst. Mat. Akad. Nauk SSSR (1991). [3] M. G. Peretyat’kin, ”Semantically universal classes of models,” Algebra Logika,30, No. 4, 414–431 (1991). [4] C. C. Chang and H. J. Keisler, Model Theory, 2nd edn., North Holland (1977). · Zbl 0423.03041 [5] H. Rogers, Theory of Recursive Functions and Effective Computability, MIT Press (1987). · Zbl 0183.01401 [6] R. I. Soare, Recursively Enumerable Sets and Degrees, Springer-Verlag (1987). · Zbl 0667.03030 [7] Yu. L. Ershov, Problems of Decidability and Constructive Models [in Russian], Nauka, Moscow, (1980). [8] J. Shoenfield, Mathematical Logic [Russian translation], Nauka, Moscow (1975).
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