Journal of Symbolic Logic

Bounding homogeneous models

Barbara F. Csima, Valentina S. Harizanov, Denis R. Hirschfeldt, and Robert I. Soare

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A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model 𝒜, i.e., the elementary diagram De(𝒜) has degree 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.

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J. Symbolic Logic, Volume 72, Issue 1 (2007), 305-323.

First available in Project Euclid: 23 March 2007

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Csima, Barbara F.; Harizanov, Valentina S.; Hirschfeldt, Denis R.; Soare, Robert I. Bounding homogeneous models. J. Symbolic Logic 72 (2007), no. 1, 305--323. doi:10.2178/jsl/1174668397.

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