SCOTT COMPLEXITY OF COUNTABLE STRUCTURES

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 /a, IL, or d /a. We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is /,1+1 for 7 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 k,c,x_pi and d k,cx i. There are three other possible Scott complexities for a computable structure of high Scott rank: II.Fx, +1,. Examples of these were already known. Our examples are computable structures of Scott rank cwt-ic + 1 which, after naming finitely many constants, have Scott rank mi7c. The existence of such structures was an open question.