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

by Denis R. Hirschfeldt, Karen Lange, and Richard A. Shore

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.