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2007 Computability of Homogeneous Models
Karen Lange, Robert I. Soare
Notre Dame J. Formal Logic 48(1): 143-170 (2007). DOI: 10.1305/ndjfl/1172787551

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 (CD) theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for (Turing) degrees d0′. 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 $\cal A$ 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 $\cal A$ 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 (the degrees of complete extensions of Peano Arithmetic).

Citation

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Karen Lange. Robert I. Soare. "Computability of Homogeneous Models." Notre Dame J. Formal Logic 48 (1) 143 - 170, 2007. https://doi.org/10.1305/ndjfl/1172787551

Information

Published: 2007
First available in Project Euclid: 1 March 2007

zbMATH: 1123.03027
MathSciNet: MR2289903
Digital Object Identifier: 10.1305/ndjfl/1172787551

Subjects:
Primary: 03D30, 03D35, 03D80

Rights: Copyright © 2007 University of Notre Dame

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Vol.48 • No. 1 • 2007
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