#### Department

Mathematics

#### Document Type

Article

#### Publication Date

2015

#### 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}.

#### Recommended Citation

Induction, bounding, weak combinatorial principles, and the Homogeneous Model Theorem, D. Hirschfeldt, K. Lange, and R. Shore, To appear in Memoirs of the American Mathematical Society.

## Comments

This is the final accepted version of the following article:

Induction, bounding, weak combinatorial principles, and the Homogeneous Model Theorem, D. Hirschfeldt, K.Lange, and R. Shore, To appear in

Memoirs of the American Mathematical Society.