##
Bounding Homogeneous Models

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
*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.

drh@math.uchicago.edu