Abstract
Ajtai’s completeness theorem roughly states that a countable structure A coded in a model of arithmetic can be end-extended and expanded to a model of a given theory G if and only if a contradiction cannot be derived by a (possibly nonstandard) proof from G plus the diagram of A, provided that the proof is definable in A and contains only formulas of a standard length. The existence of such model extensions is closely related to questions in complexity theory. In this paper we give a new proof of Ajtai’s theorem using basic techniques of model theory.
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References
Ajtai M.: \({\Sigma^1_1}\) -formulae on finite structures. Ann. Pure Appl. Log. 24, 1–48 (1983)
Ajtai, M.: The complexity of the pigeonhole principle. In: Proceedings of the IEEE 29th Annual Symposium on Foundation of Computer Science, pp. 346–355 (1988)
Ajtai, M.: Generalizations of the compactness theorem and Gödel’s completeness theorem for nonstandard finite structures. In: Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, pp. 13–33 (2007)
Ajtai, M.: A Generalization of Gödel’s Completeness Theorem for Nonstandard Finite Structures, manuscript (2011)
Barwise J., Schlipf J.: An introduction to recursively saturated and resplendent models. J. Symbol. Log. 41, 531–536 (1976)
Furst M., Saxe J., Sipser M.: Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17, 13–27 (1984)
Hájek P., Pudlák P.: Metamathematics of first order arithmetic. Springer, Berlin (1993)
Kaye R.: Models of Peano Arithmetic, Oxford Logic Guides 15. Oxford University Press, Oxford, New York (1991)
Kleene S.C.: Introduction to Metamathematics. D. Van Nostrand Co., Inc., New York, NY (1952)
Krajíček J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press, Cambridge (1995)
Paris, J., Dimitracopoulos, C.: Truth definitions for Δ0 formulae. In: Logic and Algorithmic, L’Enseignement Mathematique Monographie no 30, Geneva, pp. 317–330 (1982)
Ressayre J.P.: Models with compactness properties relative to an admissible language. Ann. Pure Appl. Log. 11, 31–55 (1977)
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Supported by grant GAUK 5732/2012 and in part by grant IAA100190902 of GA AV ČR. A part of this research has been done while the author was a visiting fellow at the Isaac Newton Institute in Cambridge (programme Semantics and Syntax) in Spring 2012.
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Garlík, M. A new proof of Ajtai’s completeness theorem for nonstandard finite structures. Arch. Math. Logic 54, 413–424 (2015). https://doi.org/10.1007/s00153-014-0416-5
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DOI: https://doi.org/10.1007/s00153-014-0416-5