  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to give a proof of an unpublished theorem of Leo Harrington from the 1970's,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 $\alpha$.

Logic