Shelah's Main Gap and the generalized Borel-reducibility

2023/08/15 by Miguel Moreno, Moreno, Miguel · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2308.07510

openalex publication_date 2023/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We answer one of the main questions in generalized descriptive set theory, the Friedman-Hyttinen-Kulikov conjecture on the Borel reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel reducibility notions of complexity. For any κ satisfying κ=λ⁺=2^λ and 2^\mathfrakc≤λ=λ^ω₁, we show that if T is a classifiable theory and T' is a non-classifiable theory, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, T, the isomorphism of models of T is either analytic co-analytic, or analytically-complete.

Cited by

• On Borel sets in ideal topologies

Related

• A Borel-reducibility Counterpart of Shelah's Main Gap Theorem
• A descriptive Main Gap Theorem
• Generalized Descriptive Set Theory and Classification Theory
• On the reducibility of isomorphism relations
• Borel Complexity and the Schröder-Bernstein Property