Bounding Homogeneous Models

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

Status: published in the Journal of Symbolic Logic 72 (2007) 305 - 323.

Availability: journal version and preprint

Abstract. 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.