Randomizations of scattered sentences

In 1970, Morley introduced the notion of a sentence φ of the infinitary logic L_w1,w being scattered. He showed that if φ is scattered, then the class I(φ) of isomorphism types of countable models of φ has cardinality at most N₁, and if φ is not scattered, then I(φ) has cardinality continuum. The absolute form of Vaught’s conjecture for φ says that if φ is scattered, then I(φ) is countable. Generalizing previous work of Ben Yaacov and the author, we introduce here the notion of a separable randomization of φ, which is a separable continuous structure whose elements are random elements of countable models of φ. We improve a result by Andrews and the author, showing that if I(φ) is countable, then φ has few separable randomizations, that is, every separable randomization of φ is isomorphic to a very simple structure called a basic randomization. We also show that if φ has few separable randomizations, then φ is scattered. Hence if the absolute Vaught conjecture holds for φ, then φ has few separable randomizations if and only if I(φ) is countable, and also if and only if φ is scattered. Moreover, assuming Martin’s axiom for Ki, we show that if φ is scattered, then φ has few separable randomizations.