Hanf number for Scott sentences of computable structures

Abstract

The Hanf number for a set S of sentences in Lω 1 ,ω (or some other logic) is the least infinite cardinal κ such that for all ϕ ∈ 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 ω CK 1 . The same argument proves that ω CK 1 is the Hanf number for Scott sentences of hyperarithmetical structures.