# Blog Archives

# Topic Archive: linear orders

# Computable categoricity, linear orders and permitting

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 Δ^{0}_{2}-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 Δ^{0}_{2} degrees.

# Scott ranks of models of a theory

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}+1