The isomorphism relation between countable models and definable equivalence relations
colorful horizontal rule

Su Gao

Abstract

In this dissertation we investigate the isomorphism relation between countable models and some other definable equivalence relations. These equivalence relations are either Borel or understood in the context of orbit equivalence relations of Polish group actions. In Chapter 1 we introduce the background and terminology. In Chapter 2 we study the Polish groups with admit complete left-invariant metrics. A characterization of this class of groups is given in model theoretic terms for the category of closed subgroups of the infinite symmetric group, or the automorphism groups of countable models. In Chapter 3 we prove some results about the homeomorphism groups of compact Polish spaces and their actions. In Chapter 4-6 various dichotomy theorems are proved for the isomorphism relation on invariant Borel classes of linear orderings, simple trees and models in the language of a single unary function. Corollaries of these theorems to answer some questions of Friedman and Stanley are presented in Chapter 7, together with a survey of other positive and negative results about the notion of FS-reducibility and strong FS-reducibility. Chapter 8 deals with a separate topic. In this chapter we study the well-known O notation and its relation with some other Borel equivalence relations arising from analysis and set theory.

Table of Contents

  1. Introduction 1.1 Basic concepts
    1.2 Historical development
    1.3 Logic actions
    1.4 Model theory of infinitary logic
  2. Polish Groups with Complete Left-Invariant Metrics 2.1 Background
    2.2 Closure under group extensions
    2.3 Homeomorphism groups of compact Polish spaces
    2.4 Automorphism groups of countable structures
  3. Actions of Homeomorphism Groups of Compact Polish Spaces 3.1 Universal spaces
    3.2 Remarks on H([0,1])
    3.3 Action by conjugacy on C(X,X)
  4. Dichtomy Theorems for Linear Orderings 4.1 An overview
    4.2 The Glimm-Effros dichotomy
  5. Dichotomy Theorems for Simple Trees 5.1 Scott ranks and automorphisms
    5.2 A strong dichotomy theorem
    5.3 Corollaries and remarks
  6. Dichotomy Theorems for Models with One Unary Function 6.1 The strong dichotomy
    6.2 Remarks on first-order theories
  7. FS-Reducibility Theory 7.1 Interpretations between L1 theories
    7.2 An application of the Glimm-Effros dichotomies
    7.3 Borel isomorphism relations among simple trees
    7.4 Borel isomorphism relations among linear orderings
  8. The O Notation and Related Equivalence Relations 8.1 Background and definitions
    8.2 \Theta and l^\infty
    8.3 More about l^\infty
    Bibliography

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This disseration was submitted to UCLA in 1998.

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