Remmel showed that a computable linear order L is computably categorical if and only if the order type of L has only a finite number of successivities. As part of the proof, Remmel assumes that L has infinitely many successivities and constructs another computable linear order R, which is not computably isomorphic to L, and a Δ⁰₂-isomorphism f such that f is an isomorphism from L onto R. Hence showing that L is not computably categorical. In this talk I will discuss the conditions under which we can use permitting arguments to construct f below certain types of Δ⁰₂ degrees.

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 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 ω₁^CK+1 which is not approximated by models of low Scott rank.