Bounding Homogeneous Models
Status: published in the Journal
of Symbolic Logic 72 (2007)
305 - 323.
version and preprint
Abstract. A Turing degree d is homogeneous
if every complete decidable (CD) theory has a d-decidable
homogeneous model A, i.e., the elementary diagram
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.