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Definability and Scott rank in separable Metric structures

Diego Bejarano

Abstract

We give a notion of Scott rank for separable metric structures based on the definability of the (metric closures of) automorphism orbits in continuous infinitary logic. This is a continuous analogue of work of Montalbán for countable structures. In the process, we prove some results concerning definability, type omitting, and back-and-forth for metric structures.

Definability and Scott rank in separable Metric structures

Abstract

We give a notion of Scott rank for separable metric structures based on the definability of the (metric closures of) automorphism orbits in continuous infinitary logic. This is a continuous analogue of work of Montalbán for countable structures. In the process, we prove some results concerning definability, type omitting, and back-and-forth for metric structures.

Paper Structure

This paper contains 13 sections, 1 theorem, 89 equations.

Table of Contents

  1. Introduction
  2. Continuous Infinitary Logic
  3. Weak Modulus of Continuity
  4. Quantifier complexity
  5. Definable Predicates
  6. Definable Sets
  7. Types
  8. Model Existence & Type Omitting Theorems
  9. The Model Existence Theorem
  10. The Type Omitting Theorem
  11. Back and Forth
  12. Definability, Scott Ranks, and Scott Predicates
  13. A Robust Scott Rank For Separable Structures

Key Result

Theorem 1

Let ${\bar{\mathcal{A}}}$ be a separable metric structure, $n$ a positive integer, and $\bar{a}\in{\bar{\mathcal{A}}}^{n}$:

Theorems & Definitions (34)

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  • ...and 24 more