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

TL;DR

This paper demonstrates that extending Morley's theorem to second-order logic leads to undecidability and explores the classification of equivalence relations in set theory models using advanced set-theoretic techniques.

Contribution

It introduces the undecidability of Morley's theorem extension to second-order logic and analyzes equivalence classes in set theory models with sophisticated methods.

Findings

Morley's theorem extension is undecidable in second-order logic

Classifies equivalence classes of $\sigma$-projective relations in set theory models

Employs forcing, Woodin cardinals, and Inner Model Theory techniques

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 $\sigma$-projective equivalence relations in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.