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On unsuperstable theories in GDST

Miguel Moreno

TL;DR

The work addresses the descriptive-set-theoretic complexity of isomorphism relations for theories in the Generalized Baire space by developing a colored-tree framework that encodes ω-equivalence into isomorphism types. It constructs ordered colored trees from κ-colorable linear orders and lifts these encodings into Generalized Ehrenfeucht–Mostowski models to realize unsuperstable theories, establishing a continuous reduction from $=^2_ω$ to isomorphism for such theories. Under suitable cardinality and forcing assumptions, the paper extends HKM results to the unsuperstable regime and derives Σ^1_1-completeness results, linking model-theoretic instability with high descriptive-set-theoretic complexity. The approach provides a unifying method to compare classifiable and unsuperstable theories in GDST and yields a robust framework for analyzing the complexity of isomorphism relations via tree-based skeletons and EM-models.

Abstract

We study the $κ$-Borel-reducibility of isomorphism relations of complete first order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T', if T is classifiable and T' is unsuperstable, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to $κ$-Borel-reducibility.

On unsuperstable theories in GDST

TL;DR

The work addresses the descriptive-set-theoretic complexity of isomorphism relations for theories in the Generalized Baire space by developing a colored-tree framework that encodes ω-equivalence into isomorphism types. It constructs ordered colored trees from κ-colorable linear orders and lifts these encodings into Generalized Ehrenfeucht–Mostowski models to realize unsuperstable theories, establishing a continuous reduction from to isomorphism for such theories. Under suitable cardinality and forcing assumptions, the paper extends HKM results to the unsuperstable regime and derives Σ^1_1-completeness results, linking model-theoretic instability with high descriptive-set-theoretic complexity. The approach provides a unifying method to compare classifiable and unsuperstable theories in GDST and yields a robust framework for analyzing the complexity of isomorphism relations via tree-based skeletons and EM-models.

Abstract

We study the -Borel-reducibility of isomorphism relations of complete first order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T', if T is classifiable and T' is unsuperstable, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to -Borel-reducibility.
Paper Structure (12 sections, 18 theorems, 60 equations)

This paper contains 12 sections, 18 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ordered trees
  4. Background
  5. Colorable orders
  6. Ordered Colored Trees
  7. Colored trees
  8. Construction of ordered coloured trees
  9. The Models
  10. Generalized Ehrenfeucht-Mostowski models
  11. Reduction of the Isomorphism Relation
  12. Corollaries

Key Result

Lemma 2.12

There is a $\kappa$-representation $\langle I^0_\alpha\mathrel{|}\alpha<\kappa\rangle$ such that for all limit $\delta<\kappa$ and $\nu\in I^0$ there is $\beta<\delta$ which satisfies the following:

Theorems & Definitions (79)

  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2: $\kappa$-representation
  • Definition 2.3: $CUB$-invariant
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 69 more