On the Theories of McDuff’s II$_1$ Factors

Recently, Boutonnet, Chifan, and Ioana proved that McDuff’s family of continuum many pairwise non-isomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two distinct members of the family are non-isomorphic. We use Ehrenfeucht–Fraisse games to provide an upper bound on the quantifier-depth of sentences which distinguish these theories.