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Generalized Descriptive Set Theory and Classification Theory

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov

TL;DR

This work extends descriptive set theory to uncountable contexts by working in generalized Baire spaces $\kappa^\kappa$ (and $2^\kappa$) under $\kappa^{<\kappa}=\kappa$, defining generalized Borel, $\Sigma_1^1$, $\Pi_1^1$, and $\Delta_1^1$ classes, and linking them to model-theoretic complexity via isomorphism relations $\cong_T^\kappa$. It develops EF-game-based and tree-coding techniques to connect stability-theoretic classification (classifiable vs unstable, DOP/OTOP, shallow) with the definability and reducibility of isomorphism relations, proving that classifiable shallow theories yield Borel (often $\Delta_1^1$) isomorphism relations, while unstable or non-classifiable theories tend to be non-Borel or non-$\Delta_1^1$. The paper also generalizes Vaught’s theorem, investigates Silver-type dichotomies, and analyzes regularity properties of the CUB filter and equivalence modulo the non-stationary ideal, including antichains and cofinality interplays. Collectively, it provides a model-theoretically natural framework for comparing the descriptive-set-theoretic complexity of isomorphism relations in uncountable settings, with forcing arguments clarifying when certain dichotomies persist or fail. These results bridge stability theory and uncountable descriptive set theory, offering tools to classify uncountable theories via the complexity of their isomorphism relations.

Abstract

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.

Generalized Descriptive Set Theory and Classification Theory

TL;DR

This work extends descriptive set theory to uncountable contexts by working in generalized Baire spaces (and ) under , defining generalized Borel, , , and classes, and linking them to model-theoretic complexity via isomorphism relations . It develops EF-game-based and tree-coding techniques to connect stability-theoretic classification (classifiable vs unstable, DOP/OTOP, shallow) with the definability and reducibility of isomorphism relations, proving that classifiable shallow theories yield Borel (often ) isomorphism relations, while unstable or non-classifiable theories tend to be non-Borel or non-. The paper also generalizes Vaught’s theorem, investigates Silver-type dichotomies, and analyzes regularity properties of the CUB filter and equivalence modulo the non-stationary ideal, including antichains and cofinality interplays. Collectively, it provides a model-theoretically natural framework for comparing the descriptive-set-theoretic complexity of isomorphism relations in uncountable settings, with forcing arguments clarifying when certain dichotomies persist or fail. These results bridge stability theory and uncountable descriptive set theory, offering tools to classify uncountable theories via the complexity of their isomorphism relations.

Abstract

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.

Paper Structure

This paper contains 39 sections, 58 theorems, 244 equations.

Table of Contents

  1. History and Motivation
  2. Introduction
  3. Notations and Conventions
  4. Set Theory
  5. Functions
  6. Model Theory
  7. Reductions
  8. Ground Work
  9. Trees and Topologies
  10. Ehrenfeucht-Fraïssé Games
  11. Coding Models
  12. Coding Partial Isomorphisms
  13. Generalized Borel Sets
  14. Borel Sets, ${\Delta_1^1}$ Sets and Infinitary Logic
  15. The Language $L_{\kappa^+\kappa}$ and Borel Sets
  16. ...and 24 more sections

Key Result

Theorem 4

For $n<\omega$ the spaces $(\kappa^\kappa)^n$ and $\kappa^\kappa$ are homeomorphic. 3

Theorems & Definitions (97)

  • Definition 1
  • Example 2
  • Definition 3
  • Theorem 4
  • Remark
  • Definition 5: Ehrenfeucht-Fraïssé games
  • Remark
  • Theorem 6: $\kappa^{<\kappa}=\kappa$
  • Remark
  • Remark
  • ...and 87 more