Scott complexity of countable structures

Abstract

We define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is, or. We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is for a limit ordinal. This answers a question left open by A. Miller. We also construct examples of computable structures of high Scott rank with Scott complexities and. There are three other possible Scott complexities for a computable structure of high Scott rank:,. Examples of these were already known. Our examples are computable structures of Scott rank which, after naming finitely many constants, have Scott rank. The existence of such structures was an open question.