Hanf Number for Scott Sentences of Computable Structures

The Hanf number for a set $S$ of sentences in $L_{ω_1,ω}$ (or some other logic) is the least infinite cardinal $κ$ such that for all $φ\in S$, if $φ$ has models in all infinite cardinalities less than $κ$, then it has models of all infinite cardinalities. S-D. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is $\beth_{ω_1^{CK}}$. The same argument proves that $\beth_{ω_1^{CK}}$ is the Hanf number for Scott sentences of hyperarithmetical structures.

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