On the computability of optimal Scott sentences
Abstract
Given a countable mathematical structure, its Scott sentence is a sentence of the infinitary logic Lω1 ω that characterizes it among all countable structures. We can measure the complexity of a structure by the least complexity of a Scott sentence for that structure. It is known that there can be a difference between the least complexity of a Scott sentence and the least complexity of a computable Scott sentence; for example, Alvir, Knight, and McCoy showed that there is a computable structure with a 2 Scott sentence but no computable 2 Scott sentence. It is well known that a structure with a 2 Scott sentence must have a computable 4 Scott sentence. We show that this is best possible: there is a computable structure with a 2 Scott sentence but no computable 4 Scott sentence. We also show that there is no reasonable characterization of the computable structures with a computable n Scott sentence by showing that the index set of such structures is 11-m-complete.
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