# Scattered Sentences have Few Separable Randomizations

## TL;DR

This paper proves that under ZFC, every scattered sentence has few separable randomizations, confirming a conjecture and linking it to the absolute Vaught conjecture, thus advancing understanding in model theory.

## Contribution

It demonstrates that the conclusion about scattered sentences and separable randomizations can be proved in ZFC alone, removing the need for Martin's axiom.

## Key findings

- Proved that scattered sentences have few separable randomizations in ZFC.
- Established the equivalence between the absolute Vaught conjecture and properties of $L_{oldsymbol{
u}}$-sentences.
- Connected properties of scattered sentences to the countability of models.

## Abstract

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.