Submitted. PDF. arXiv. Bibtex.

 @ARTICLE{EHMT21,
   AUTHOR= {C. J. Eagle and C. Hamel and S. Müller and F. D. Tall},
   TITLE= {An undecidable extension of {Morley's theorem} on the number of countable
 models},
   Note={Submitted},
   EPRINT ={2107.07636}}

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.