On the minimality of tame models in the isols

Barback, Joseph

doi:10.1090/S0002-9939-1993-1155592-0

On the minimality of tame models in the isols
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Proc. Amer. Math. Soc. 119 (1993), 935-939 Request permission

Abstract:

Based on the work of Hirschfeld, it is known that there is a close connection between models for the $\Pi _2^0$ fragment of arithmetic and homomorphic images of the semiring of recursive functions. This fragment of arithmetic includes most of the familiar results of classical number theory. There is a realization of this fragment in the isols in systems called tame models. In this paper a new proof is given to the following result of Ellentuck and McLaughlin on the minimality of tame models: If two tame models share an infinite element, then the models are equal.

References

Joseph Barback, On tame regressive isols, Houston J. Math. 8 (1982), no. 2, 153–159. MR 674033
Joseph Barback, Tame models in the isols, Houston J. Math. 12 (1986), no. 2, 163–175. MR 862034
Joseph Barback, Corrections for two papers: “On tame regressive isols” [Houston J. Math. 8 (1982), no. 2, 153–159; MR0674033 (84b:03061)] and “Tame models in the isols” [Houston J. Math. 12 (1986), no. 2, 163–175; MR0862034 (88d:03090a)], Houston J. Math. 13 (1987), no. 3, 301–302. MR 916138
J. C. E. Dekker and J. Myhill, Recursive equivalence types, Univ. California Publ. Math. 3 (1960), 67–213. MR 0117155
Erik Ellentuck, Diagonal methods in the theory of isols, Z. Math. Logik Grundlagen Math. 26 (1980), no. 3, 193–204. MR 578828, DOI 10.1002/malq.19800261302
J. Hirschfeld, Models of arithmetic and recursive functions, Israel J. Math. 20 (1975), no. 2, 111–126. MR 381969, DOI 10.1007/BF02757881
Thomas G. McLaughlin, Regressive sets and the theory of isols, Lecture Notes in Pure and Applied Mathematics, vol. 66, Marcel Dekker, Inc., New York, 1982. MR 659652
Joseph Barback, Tame models in the isols, Houston J. Math. 12 (1986), no. 2, 163–175. MR 862034
T. G. McLaughlin, Some properties of $\forall \exists$ models in the isols, Proc. Amer. Math. Soc. 97 (1986), no. 3, 495–502. MR 840636, DOI 10.1090/S0002-9939-1986-0840636-X
Thomas G. McLaughlin, Sub-arithmetical ultrapowers: a survey, Ann. Pure Appl. Logic 49 (1990), no. 2, 143–191. MR 1077076, DOI 10.1016/0168-0072(90)90064-9
Anil Nerode, Extensions to isols, Ann. of Math. (2) 73 (1961), 362–403. MR 131363, DOI 10.2307/1970338
Anil Nerode, Diophantine correct non-standard models in the isols, Ann. of Math. (2) 84 (1966), 421–432. MR 202603, DOI 10.2307/1970455