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Bounding Homogeneous Models

Status: *Journal of Symbolic Logic* vol. 72 (2007), pp. 305 - 323.

Availability: PostScript, DVI, and PDF

**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