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Models of arithmetic and upper bounds for arithmetic sets

  • Alistair H. Lachlan (a1) and Robert I. Soare (a2)

Abstract

We settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

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[1]Knight, J. F., Degrees of models with prescribed Scott set, Proceedings of the U.S.-Israel Workshop on Model Theory in Mathematical Logic: Classification Theory, Chicago, December 15–19, 1985 (John Baldwin, editor), Lecture Notes in Mathematics, vol. 1292. Springer-Verlag, Berlin. Heidelberg, and New York, 1987, pp. 182–191.
[2]Knight, J. F., A metatheorem for constructions by finitely many workers, this Journal, vol. 55 (1990), pp. 787–804.
[3]Knight, J., Lachlan, A. H., and Soare, R. I., Two Theorems On Degrees Of Models Of Arithmetic, this Journal, vol. 49 (1984), pp. 425–436.
[4]Lerman, M., Upper bounds for the arithmetical degrees, Annals of Pure and Applied Logic, vol. 29 (1985), pp. 225–253.
[5]Macintyre, A. and Marker, D., Degrees of recursively saturated models, Transactions of the American Mathematical Society, vol. 282 (1984), pp. 539–554.
[6]Marker, D., Degrees of models of true arithmetic, Proceedings of the Herbrand Symposium: Logic Colloquium, 1981 (J. Stern, editor), North-Holland, Amsterdam, 1982, pp. 233–242.
[7]Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Recursive Function Theory, Proceedings of Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1961, pp. 117–121.
[8]Solovay, R. M., Degrees of models of true arithmetic, preliminary version, unpublished manuscript (1983).

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Models of arithmetic and upper bounds for arithmetic sets

  • Alistair H. Lachlan (a1) and Robert I. Soare (a2)

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