Scott processes

Abstract

The Scott process of a relational structure $M$ is the sequence of sets of formulas given by the Scott analysis of $M$. We present axioms for the class of Scott processes of structures in a relational vocabulary $\tau$, and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970's, showing that a counterexample to Vaught's Conjecture has models of cofinally many Scott ranks below $\omega_{2}$. Our approach also gives a theorem of Harnik and Makkai, showing that if there exists a counterexample to Vaught's Conjecture, then there is a counterexample whose uncountable models have the same $\mathcal{L}_{\omega_{1}, \omega}(\tau)$-theory, and which has a model of Scott rank $\omega_{1}$. Moreover, we show that if $\phi$ is a sentence of $\mathcal{L}_{\omega_{1}, \omega}(\tau)$ giving rise to a counterexample to Vaught's Conjecture, then for every limit ordinal $\alpha$ greater than the quantifier depth of $\phi$ and below $\omega_{2}$, $\phi$ has a model of Scott rank $\alpha$.