Hanf Number for Scott Sentences of Computable Structures

Abstract

The Hanf number for a set $S$ of sentences in $L_{{\omega }_1,\omega }$ (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. S-D. 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.