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 ACA0, IΣ02, and
BΣ02. Others, however,
exhibit complex interactions with first order induction and bounding
principles. In particular, we isolate several principles that are
provable from IΣ02, are (more than)
arithmetically
conservative over RCA0, and imply
IΣ02 over
BΣ02. In an attempt to capture the
combinatorics of this
class of principles, we introduce the principle
Π01GA, as well
as its generalization Π0nGA, which is
conservative over RCA0
and equivalent to IΣ0n+1 over
BΣ0n+1.
drh@math.uchicago.edu