Paper detail

Scott processes

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for the class of Scott processes of structures in a relational vocabulary $τ$, and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970's, showing that a counterexample to Vaught's Conjecture has models of cofinally many Scott ranks below $ω_{2}$. 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 $\mathcal{L}_{ω_{1}, ω}(τ)$-theory, and which has a model of Scott rank $ω_{1}$. Moreover, we show that if $ϕ$ is a sentence of $\mathcal{L}_{ω_{1}, ω}(τ)$ giving rise to a counterexample to Vaught's Conjecture, then for every limit ordinal $α$ greater than the quantifier depth of $ϕ$ and below $ω_{2}$, $ϕ$ has a model of Scott rank $α$.