Hanf Number for Scott Sentences of Computable Structures

TL;DR

This paper determines that the Hanf number for Scott sentences of computable and hyperarithmetical structures is the Beth number _{\u03c9_1^{CK}}, establishing a precise cardinality threshold for the existence of models.

Contribution

It establishes the exact Hanf number for Scott sentences of computable and hyperarithmetical structures as _{\u03c9_1^{CK}}, answering a question posed by S-D. Friedman.

Findings

Hanf number for Scott sentences is _{\u03c9_1^{CK}}.

The same Hanf number applies to hyperarithmetical structures.

Provides a precise cardinality threshold for model existence.

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 $\varphi\in S$, if $\varphi$ 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.