Annals of Pure and Applied Logic. Volume 174, Issue 9, October-November 2023. 103317.
DOI: 10.1016/j.apal.2023.103317. PDF. arXiv. Bibtex.

 @ARTICLE{EHMT23,
   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},
   JOURNAL={Annals of Pure and Applied Logic},
   EPRINT ={2107.07636},
   VOLUME = {174},
   ISSUE = {9},
   YEAR = {2023},
   PAGES = {103317},
   DOI = {10.1016/j.apal.2023.103317}}
   

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.