Scott processes
Paul Larson · 2017
The Scott process of a relational structure M is the transfinite sequence of sets of formulas given by the Scott analysis of M , as introduced in [ Sco65 ]. We present axioms for the class of Scott processes of infinite structures in a relational vocabulary τ, and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970s, showing that a counterexample to Vaught’s Conjecture has models of cofinally many Scott ranks below ω 2 .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 all have the same LN 1 , 0 (τ)-theory, and which has a model of Scott rank ω 1 . Moreover, we show that if ϕ is a sentence of LN 1 , 0 (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 , α,ϕ has a model of Scott rank ϕ, and that for club many ordinals α below each of ω 1 and ω 2 , α,ϕ has at least two nonisomorphic models of Scott rank α, generalizing a result of Sacks. We give a new proof using Scott processes of the fact that if there is a counterexample to Vaught’s Conjecture in LN 1 , 0 , then there is one of quantifier depth ω. We show that Scott processes give rise to a class of structures with the property if there is a counterexample to Vaught’s Conjecture then there is one corresponding to a subclass of this class. We show that if countable structures M and N have the same Scott process though level δ, for δ a countable ordinal, and N has Scott rank δ, then M is isomorphic to a quantifier-depth-δ-elementary submodel of N .