Computable trees, prime models, and relative decidability

Hirschfeldt, Denis

doi:10.1090/S0002-9939-05-08097-4

Computable trees, prime models, and relative decidability
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Proc. Amer. Math. Soc. 134 (2006), 1495-1498 Request permission

Abstract:

We show that for every computable tree $\mathcal T$ with no dead ends and all paths computable, and every $D >_T \emptyset$, there is a $D$-computable listing of the isolated paths of $\mathcal T$. It follows that for every complete decidable theory $T$ such that all the types of $T$ are computable and every $D >_T \emptyset$, there is a $D$-decidable prime model of $T$. This result extends a theorem of Csima and yields a stronger version of the theorem, due independently to Slaman and Wehner, that there is a structure with presentations of every nonzero degree but no computable presentation.

References

Barbara F. Csima, Degree spectra of prime models, J. Symbolic Logic 69 (2004), no. 2, 430–442. MR 2058182, DOI 10.2178/jsl/1082418536
B. F. Csima, D. R. Hirschfeldt, V. S. Harizanov, and R. I. Soare, Bounding homogeneous models, to appear.
Barbara F. Csima, Denis R. Hirschfeldt, Julia F. Knight, and Robert I. Soare, Bounding prime models, J. Symbolic Logic 69 (2004), no. 4, 1117–1142. MR 2135658, DOI 10.2178/jsl/1102022214
S. S. Gončarov and A. T. Nurtazin, Constructive models of complete decidable theories, Algebra i Logika 12 (1973), 125–142, 243 (Russian). MR 0398816
Valentina S. Harizanov, Pure computable model theory, Handbook of recursive mathematics, Vol. 1, Stud. Logic Found. Math., vol. 138, North-Holland, Amsterdam, 1998, pp. 3–114. MR 1673621, DOI 10.1016/S0049-237X(98)80002-5
Leo Harrington, Recursively presentable prime models, J. Symbolic Logic 39 (1974), 305–309. MR 351804, DOI 10.2307/2272643
Terrence S. Millar, Foundations of recursive model theory, Ann. Math. Logic 13 (1978), no. 1, 45–72. MR 482430, DOI 10.1016/0003-4843(78)90030-X
Theodore A. Slaman, Relative to any nonrecursive set, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2117–2122. MR 1443408, DOI 10.1090/S0002-9939-98-04307-X
Stephan Wehner, Enumerations, countable structures and Turing degrees, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2131–2139. MR 1443415, DOI 10.1090/S0002-9939-98-04314-7