##
Induction, Bounding, Weak Combinatorial Principles, and the
Homogeneous Model Theorem

Status: published in the *Memoirs
of the American Mathematical Society* 249 (2017),
no. 1187, 101 pp.

Availability: journal
version and preprint

**Abstract.** Goncharov and Peretyat'kin independently gave necessary
and sufficient
conditions for when a set of types of a complete theory *T* is the
type spectrum of some homogeneous model of *T*. Their result can be
stated as a principle of second order arithmetic, which we call the
Homogeneous Model Theorem (HMT), and analyzed from the points of view
of computability theory and reverse mathematics. Previous
computability theoretic results by Lange suggested a close connection
between HMT and the Atomic Model Theorem (AMT), which states that
every complete atomic theory has an atomic model. We show that HMT and
AMT are indeed equivalent in the sense of reverse mathematics, as well
as in a strong computability theoretic sense. We do the same for an
analogous result of Peretyat'kin giving necessary and sufficient
conditions for when a set of types is the type spectrum of some
model.

Along the way, we analyze a number of related principles. Some of
these turn out to fall into well-known reverse mathematical classes,
such as ACA_{0}, IΣ^{0}_{2}, and
BΣ^{0}_{2}. Others, however,
exhibit complex interactions with first order induction and bounding
principles. In particular, we isolate several principles that are
provable from IΣ^{0}_{2}, are (more than)
arithmetically
conservative over RCA_{0}, and imply
IΣ^{0}_{2} over
BΣ^{0}_{2}. In an attempt to capture the
combinatorics of this
class of principles, we introduce the principle
Π^{0}_{1}GA, as well
as its generalization Π^{0}_{n}GA, which is
conservative over RCA_{0}
and equivalent to IΣ^{0}_{n+1} over
BΣ^{0}_{n+1}.

drh@math.uchicago.edu