# Scott ranks of models of a theory

## Matthew Harrison-Trainor

### University of California Berkeley

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 *L _{ω1ω}*.) 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 *Π _{2}^{0}* 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

*ω*which is not approximated by models of low Scott rank.

_{1}^{CK}+1Matthew Harrison-Trainor is a doctoral student at the University of California at Berkeley, studying computability, computable model theory, and model theory. His advisor is Prof. Antonio Montalban.