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Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem
 
Denis R. Hirschfeldt :  University of Chicago, Chicago, Illinois, USA
Karen Lange :  Wellesley College, Wellesley, Massachusetts, USA
Richard A. Shore :  Cornell University, Ithaca, New York, USA
Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem
eBook ISBN:  978-1-4704-4141-8
Product Code:  MEMO/249/1187.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
Add to cart
Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem
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Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem
Denis R. Hirschfeldt :  University of Chicago, Chicago, Illinois, USA
Karen Lange :  Wellesley College, Wellesley, Massachusetts, USA
Richard A. Shore :  Cornell University, Ithaca, New York, USA
eBook ISBN:  978-1-4704-4141-8
Product Code:  MEMO/249/1187.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
Add to cart
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 249; 2017; 101 pp
    MSC: Primary 03

    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 is called 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. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and 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.
  • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Memoirs of the American Mathematical Society
Volume: 249; 2017; 101 pp
MSC: Primary 03

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 is called 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. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and 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.
  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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