Scattered Sentences have Few Separable Randomizations

In the paper “Randomizations of Scattered Sentences”, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every $L_{{\omega }_1\omega }$-sentence with few separable randomizations has countably many countable models.