Topic Archive: infinitary formulas
I will talk about a few different results about the Scott ranks of models of a theory. (By a theory, I mean a sentence of Lω1ω.) These results are all related in that they all follow from the same general construction; this construction takes a pseudo-elementary class C of linear orders and produces a theory T such that the Scott ranks of models of T are related to the well-founded parts of linear orders in C.
The main result is a descriptive-set-theoretic classification of the collections of ordinals which are the Scott spectrum of a theory. We also answer some open questions. First, we show that for each ordinal β, there is a Π20 theory which has no models of Scott rank less than β. Second, we find the Scott height of computable infinitary sentences. Third, we construct a computable model of Scott rank ω1CK+1 which is not approximated by models of low Scott rank.