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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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