Computability of Homogeneous Models

Notre Dame Journal of Formal Logic 48 (1):143-170 (2007)
Abstract
In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous models," which generalize the older results and which include positive results on when a certain homogeneous model of T has an isomorphic copy of a given Turing degree. We then survey Lange's "A characterization of the 0-basis homogeneous bounding degrees" for negative results about when does not have such copies, generalizing negative results by Goncharov, Peretyat'kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare, "Bounding homogeneous models," about degrees d that are homogeneous bounding and explain their relation to the PA degrees
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Karen Lange (2008). The Degree Spectra of Homogeneous Models. Journal of Symbolic Logic 73 (3):1009-1028.
Robert I. Soare (1996). Computability and Recursion. Bulletin of Symbolic Logic 2 (3):284-321.
Libo Lo (1983). On the Number of Countable Homogeneous Models. Journal of Symbolic Logic 48 (3):539-541.
Stewart Shapiro (1983). Remarks on the Development of Computability. History and Philosophy of Logic 4 (1-2):203-220.
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