Notre Dame Journal of Formal Logic

Computability of Homogeneous Models

Karen Lange and Robert I. Soare

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).

Article information

Source
Notre Dame J. Formal Logic, Volume 48, Number 1 (2007), 143-170.

Dates
First available in Project Euclid: 1 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1172787551

Digital Object Identifier
doi:10.1305/ndjfl/1172787551

Mathematical Reviews number (MathSciNet)
MR2289903

Zentralblatt MATH identifier
1123.03027

Subjects
Primary: 03D35: Undecidability and degrees of sets of sentences 03D30: Other degrees and reducibilities 03D80: Applications of computability and recursion theory

Keywords
prime saturated homogeneous model computable model theory compatability theory

Citation

Lange, Karen; Soare, Robert I. Computability of Homogeneous Models. Notre Dame J. Formal Logic 48 (2007), no. 1, 143--170. doi:10.1305/ndjfl/1172787551. https://projecteuclid.org/euclid.ndjfl/1172787551


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