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

About this Title

Denis R. Hirschfeldt, Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637, Karen Lange, Department of Mathematics, Wellesley College, 106 Central St., Wellesley, Massachusetts 02481 and Richard A. Shore, Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 249, Number 1187
ISBNs: 978-1-4704-2657-6 (print); 978-1-4704-4141-8 (online)
DOI: https://doi.org/10.1090/memo/1187
Published electronically: August 9, 2017
Keywords: Reverse mathematics, computable model theory, atomic models, and homogeneous models.
MSC: Primary 03B30; Secondary 03C07, 03C15, 03C50, 03C57, 03D45, 03F30, 03F35

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Definitions
  • 3. The Atomic Model Theorem and Related Principles
  • 4. Defining Homogeneity
  • 5. Closure Conditions and Model Existence
  • 6. Extension Functions and Model Existence
  • 7. The Reverse Mathematics of Model Existence Theorems
  • 8. Open Questions
  • A. Approximating Generics
  • B. Atomic Trees
  • C. Saturated Models

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$_0$, I$\Sigma ^0_2$, and B$\Sigma ^0_2$. Others, however, exhibit complex interactions with first order induction and bounding principles. In particular, we isolate several principles that are provable from I$\Sigma ^0_2$, are (more than) arithmetically conservative over RCA$_0$, and imply I$\Sigma ^0_2$ over B$\Sigma ^0_2$. In an attempt to capture the combinatorics of this class of principles, we introduce the principle $\Pi ^0_1$GA, as well as its generalization $\Pi ^0_n$GA, which is conservative over RCA$_0$ and equivalent to I$\Sigma ^0_{n+1}$ over B$\Sigma ^0_{n+1}$.

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