Annals of Pure and Applied Logic. Volume 174, Issue 9, October-November 2023. 103317.

DOI: 10.1016/j.apal.2023.103317. PDF. arXiv.
Bibtex.

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.