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Definability in affine continuous logic

Seyed-Mohammad Bagheri

Abstract

I study definable sets in affine continuous logic. Let $T$ be an affine theory. After giving some general results, it is proved that if $T$ has a first order model, its extremal theory is a complete first order theory and first order definable sets are affinely definable. In this case, the type spaces of $T$ are Bauer simplices and they coincide with the sets of Keisler measures of the extremal theory. In contrast, if $T$ has a compact model, definable sets are exactly the end-sets of definable predicates. As an example, it is proved in the theory of probability algebras that one dimensional definable sets are exactly the intervals $[a,b]$.

Definability in affine continuous logic

Abstract

I study definable sets in affine continuous logic. Let be an affine theory. After giving some general results, it is proved that if has a first order model, its extremal theory is a complete first order theory and first order definable sets are affinely definable. In this case, the type spaces of are Bauer simplices and they coincide with the sets of Keisler measures of the extremal theory. In contrast, if has a compact model, definable sets are exactly the end-sets of definable predicates. As an example, it is proved in the theory of probability algebras that one dimensional definable sets are exactly the intervals .
Paper Structure (10 sections, 32 theorems, 74 equations)

This paper contains 10 sections, 32 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Types
  4. Definable predicates
  5. Definable functions
  6. Definable sets
  7. Principal types
  8. Definability in first order models
  9. Definability in compact models
  10. Definability in measure algebras

Key Result

Theorem 2.1

(Ultramean theorem) For every affine formula $\phi(x_1,...x_n)$ and $[a^1_i],...,[a^n_i]$

Theorems & Definitions (60)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Theorem 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Proposition 4.1
  • ...and 50 more