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Hanf number for Scott sentences of computable structures. (English) Zbl 07001423
Summary: The Hanf number for a set \(S\) of sentences in \(\mathcal {L}_{\omega _1,\omega }\) (or some other logic) is the least infinite cardinal \(\kappa \) such that for all \(\phi \in S\), if \(\phi \) has models in all infinite cardinalities less than \(\kappa \), then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \(\beth _{\omega _1^{CK}}\). The same argument proves that \(\beth _{\omega _1^{CK}}\) is the Hanf number for Scott sentences of hyperarithmetical structures.
MSC:
03C57 Computable structure theory, computable model theory
03D45 Theory of numerations, effectively presented structures
03C75 Other infinitary logic
03C70 Logic on admissible sets
03C52 Properties of classes of models
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