Bounding homogeneous models

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 D^e(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.