# Ioannis Souldatos: The Hanf number for Scott sentences of computable structures.

Thursday, April 21, 2016, 4:00–5:30 PM, 3088 East Hall.

We will prove the following two theorems:
Theorem 1: Let A be a computable structure for a computable vocabulary \tau, and let \sigma be a Scott sentence for A. If \sigma has models of cardinality \beth_\alpha for all \alpha<\omega_1^{CK}, then it has models of all infinite cardinalities.
Theorem 2: (Using Kleene’s O:) For every ordinal notation a\in O, there exists a computable structure A, such that A characterizes \beth_{|a|}, where |a| is the ordinal defined by a.
Combining the above two theorems we obtain that the Hanf number for Scott sentences of computable structures is equal to \beth_{\omega_1^{CK}}. This answers a question of Sy D. Friedman.