Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On classification of continuous first order theories

Karim Khanaki

TL;DR

The paper develops a function-space–driven classification of continuous first-order theories, extending notions of IP and SOP to the continuous setting via epsilon-SOP and semi-uniform IP-blocking. It connects model-theoretic stability notions to Banach-space phenomena, showing that $NIP$ corresponds to convergence to $Baire ext{-}1/2$ functions and to DBSC-definability of coheirs, while SOP is tied to $c_0$-embeddings in spaces of formulas; crucially, a perfect Shelah-type theorem does not hold in continuous logic due to subtler Baire-class distinctions. By introducing a C-property-based framework and discussing $ ext{B}_1^\xi$ hierarchies, the work outlines a refined classification of continuous theories that parallels classical dividing lines while accommodating function-space intricacies. These results illuminate deep links between model theory, function theory, and Banach-space structure, suggesting new dividing lines and guiding principles for the theory of continuous logic and its applications to analysis.

Abstract

We give several new characterizations of $IP$ (the independence property) and $SOP$ (the strict order property) for continuous first order logic and study their relations to the function theory and the Banach space theory. We suggest new dividing lines of unstable theories by the study of subclasses of Baire-1 functions and argue why one should not expect a perfect analog of Shelah's theorem, namely a theory is unstable iff it has $IP$ or $SOP$, for real-valued logics, especially for continuous logic.

On classification of continuous first order theories

TL;DR

The paper develops a function-space–driven classification of continuous first-order theories, extending notions of IP and SOP to the continuous setting via epsilon-SOP and semi-uniform IP-blocking. It connects model-theoretic stability notions to Banach-space phenomena, showing that corresponds to convergence to functions and to DBSC-definability of coheirs, while SOP is tied to -embeddings in spaces of formulas; crucially, a perfect Shelah-type theorem does not hold in continuous logic due to subtler Baire-class distinctions. By introducing a C-property-based framework and discussing hierarchies, the work outlines a refined classification of continuous theories that parallels classical dividing lines while accommodating function-space intricacies. These results illuminate deep links between model theory, function theory, and Banach-space structure, suggesting new dividing lines and guiding principles for the theory of continuous logic and its applications to analysis.

Abstract

We give several new characterizations of (the independence property) and (the strict order property) for continuous first order logic and study their relations to the function theory and the Banach space theory. We suggest new dividing lines of unstable theories by the study of subclasses of Baire-1 functions and argue why one should not expect a perfect analog of Shelah's theorem, namely a theory is unstable iff it has or , for real-valued logics, especially for continuous logic.
Paper Structure (11 sections, 11 theorems, 35 equations)

This paper contains 11 sections, 11 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Subclasses of Baire 1 Functions
  4. Continuous Model Theory
  5. Strict Order Property
  6. Independence property
  7. Characterization of $NIP$
  8. Baire-1/2 Definability of Coheirs
  9. Discussion and Thesis
  10. Discussion
  11. Thesis

Key Result

Theorem 3.7

Let $T$ be a complete continuous theory. The following are equivalent: (i) $T$ has $SOP$. (ii) There are a formula $\phi(x,y)$, sequences $(a_i)$, $(b_i)$, and real numbers $r < s$ such that

Theorems & Definitions (45)

  • Definition 2.3
  • Definition 2.6
  • proof
  • Remark 2.10
  • Remark 3.1
  • Definition 3.2
  • proof
  • Definition 3.4: $SOP$ for a continuous theory
  • Remark 3.5
  • Definition 3.6
  • ...and 35 more