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


About this Title

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

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.

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Table of Contents


Chapters

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

Abstract


Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of . 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, I, and B. Others, however, exhibit complex interactions with first order induction and bounding principles. In particular, we isolate several principles that are provable from I, are (more than) arithmetically conservative over RCA, and imply I over B. In an attempt to capture the combinatorics of this class of principles, we introduce the principle GA, as well as its generalization GA, which is conservative over RCA and equivalent to I over B.

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