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Bounding homogeneous models. (English) Zbl 1116.03027

Summary: A Turing degree \({\mathbf d}\) is homogeneous bounding if every complete decidable (CD) theory has a \({\mathbf d}\)-decidable homogeneous model \({\mathcal A}\), i.e., the elementary diagram \(D^e({\mathcal A})\) has degree \({\mathbf d}\). It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory \(T\) such that every homogeneous model of \(T\) has a PA degree.

MSC:

03C57 Computable structure theory, computable model theory
03D28 Other Turing degree structures
03F30 First-order arithmetic and fragments
Full Text: DOI

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