An undecidable extension of Morley's theorem on the number of countable models

2021/07/15 by Christopher J. Eagle, Clovis Hamel, Eagle, Christopher J. +5
Computer Science · Mathematics · Psychology · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Philosophy and Theoretical Science

paper · pdf · doi:10.48550/arxiv.2107.07636

openalex publication_date 2021/07/15 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of σ-projective equivalence relations in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.

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