Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor

We show that any II ${}_1$ factor that has the same 4-quantifier theory as the hyperfinite II ${}_1$ factor $ℛ$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $ℛ$ . In the same spirit, we improve a recent result of the first-named author, who …