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axioms for the class of Scott processes of structures in a relational vocabulary τ , and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970’s, showing that a counterexample to Vaught’s Conjecture has models of cofinally many Scott ranks below ω2. 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 Lω1,ω(τ)-theory, and which has a model of Scott rank ω1. Moreover, we show that if φ is a sentence of Lω1,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 ω2, φ has a model of Scott rank α.