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ON THE COMPUTABILITY OF OPTIMAL SCOTT SENTENCES
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 17 September 2025, pp. 1-20
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- Article
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Given a countable mathematical structure, its Scott sentence is a sentence of the infinitary logic
$\mathcal {L}_{\omega _1 \omega }$ 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 
$\Pi _2$ Scott sentence but no computable 
$\Pi _2$ Scott sentence. It is well known that a structure with a 
$\Pi _2$ Scott sentence must have a computable 
$\Pi _4$ Scott sentence. We show that this is best possible: there is a computable structure with a 
$\Pi _2$ Scott sentence but no computable 
$\Sigma _4$ Scott sentence. We also show that there is no reasonable characterization of the computable structures with a computable 
$\Pi _n$ Scott sentence by showing that the index set of such structures is 
$\Pi ^1_1$-m-complete.
