Scott processes

TL;DR

This paper develops axioms for Scott processes of relational structures, providing new proofs and results related to Vaught's Conjecture, including the existence of models with various Scott ranks in counterexamples.

Contribution

It introduces axioms for Scott processes and uses them to prove the existence of models with cofinal Scott ranks below _2 in counterexamples to Vaught's Conjecture.

Findings

Counterexamples to Vaught's Conjecture have models of cofinally many Scott ranks below _2

Existence of models with Scott rank _1 in counterexamples

Counterexamples can have models of Scott rank _1 for every limit ordinal above certain depth

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for the class of Scott processes of structures in a relational vocabulary $\tau$, and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970's, showing that a counterexample to Vaught's Conjecture has models of cofinally many Scott ranks below $\omega_{2}$.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 $\alpha$.