Abstract
The notions of finitary and infinitary combinatorics were recently introduced by the author. In the present article, we discuss these notions and the corresponding semantical layers. We suggest a definition of a model-theoretical property. By author’s opinion, this definition agrees with the meaning that is generally accepted and used inmodel theory.We show that the similarity relation for theories over finitary and infinitary layers of model-theoretical properties is natural and important. Our arguments are based on comparing our approach with known model-theoretical ones.We find examples of pairs of mutually interpretable theories possessing distinct simple model-theoretical properties. These examples show weak points of the notion of mutual interpretability from the point of view of preservation of model-theoretical properties.
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References
K. L. de Bouvère, “Synonymous theories,” in The Theory of Models (North-Holland, Amsterdam, 1965), 402.
Yu. L. Ershov and S. S. Goncharov, Constructive Models (Consultants Bureau, New York, 2000) [Constructive Models (Nauchnaya Kniga, Novosibirsk, 1999)].
H. Gaifman, “Operations on relational structures, functors and classes. I,” in Proc. Tarski Sympos. (Amer. Math. Soc., Providence, RI, 1974), 21.
W. Hanf, “Model-theoretic methods in the study of elementary logic,” in The Theory of Models (North- Holland, Amsterdam, 1965), 132. Amsterdam: North-Holland Publishing Co., 1965. P. 132–145.
W. Hanf, “The Boolean algebra of logic,” Bull. Amer. Math. Soc. 81, 587 (1975).
W. Hodges, A Shorter Model Theory (Cambridge Univ. Press, Cambridge, 1997).
A. I. Mal’tsev, “Strongly related models and recursively perfect algebras,” Soviet Math., Dokl. 3, 987 (1962) [Dokl. Akad. Nauk SSSR 145, 276 (1962)].
K. L. Manders, First-Order Logical Systems and Set-Theoretical Definability (Univ. Pittsburgh, Pittsburgh, 1980) [preprint].
J. Mycielski, “A lattice of interpretability types of theories,” J. Symbolic Logic 42, 297 (1977).
J. Mycielski, P. Pudlák, and A. Stern, A Lattice of Chapters of Mathematics (Interpretations Between Theorems) (Amer.Math. Soc, Providence, RI, 1990).
D. Myers, “Lindenbaum–Tarski algebras,” in Handbook of Boolean Algebras 3 (Elsevier, Amsterdam, 1989), 1167.
D. Myers, “An interpretive isomorphism between binary and ternary relations,” in Structures in Logic and Computer Science (Springer-Verlag, Berlin, 1997), 84.
M. G. Peretyat’kin, “The similarity of properties of recursively enumerable and finitely axiomatizable theories,” Soviet Math., Dokl. 40, 372 (1990) [Dokl. Akad. Nauk SSSR 308, 788 (1989)].
M. G. Peretyat’kin, “Semantically universal classes of models,” Algebra and Logic 30, 271 (1991) [Algebra i logika 30, 414 (1991)].
M. G. Peretyat’kin, “Analogues of Rice’s theorem for semantic classes of propositions,” Algebra and Logic 30, 332 (1991) [Algebra i logika 30, 517 (1991)].
M. G. Peretyat’kin, Finitely Axiomatizable Theories (Plenum, New York, 1997) [Finitely Axiomatizable Theories (Nauchnaya Kniga, Novosibirsk, 1997)].
M. G. Peretyat’kin, “Finitely axiomatizable theories and similarity relations,” in Model Theory and Applications (Amer.Math. Soc., Providence, RI, 1999), 309.
M. G. Peretyat’kin, “On model-theoretic properties that are not preserved on the pairs of mutually interpretable theories,” in Logic Colloquium 2012–Contributed Talks (Univ. Manchester, Manchester, UK, 2012) [http://wwwclepsmanchesteracuk/medialand/maths/archivedevents/ workshops/wwwmimsmanchesteracuk/events/workshops/LC2012/abs/contribpdf].
M. G. Peretyat’kin, “Introduction in first-order combinatorics providing a conceptual framework for computation in predicate logic,” in Computation Tools (IARIA, 2013), 31 [https://wwwthinkmindorg/downloadphp?articleid=computation_tools_2013_2_10_80018].
C. C. Pinter, “Properties preserved under definitional equivalence and interpretations,” Z. Math. Logik Grundlag.Math. 24, 481 (1978).
J. R. Shoenfield, Mathematical Logic (Addison-Wesley, Reading, MA, 1967).
L. Szczerba, “Interpretability of elementary theories,” in Logic, Foundations of Mathematics and Computability Theory (D. Reidel Publishing Co., Dordrecht, 1977), 129.
W. Szmielew and A. Tarski, “Mutual interpretability of some essentially undecidable theories,” in Proc. Internat. Congr. Math. I (Amer. Math. Soc., Providence, RI, 1952), 734.
H. Rogers, Theory of Recursive Functions and Effective Computability (McGraw-Hill, New York–Toronto–London, 1967).
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Original Russian Text © M.G. Peretyat’kin, 2015, published in Matematicheskie Trudy, 2015, Vol. 18, No. 2, pp. 61–92.
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Peretyat’kin, M.G. First-order combinatorics and model-theoretical properties that can be distinct for mutually interpretable theories. Sib. Adv. Math. 26, 196–214 (2016). https://doi.org/10.3103/S1055134416030044
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DOI: https://doi.org/10.3103/S1055134416030044