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arXiv · 1602.01156

Hanf Number for Scott Sentences of Computable Structures

Sergey Goncharov·Julia Knight·Ioannis Souldatos
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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.

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Sergey Goncharov, Julia Knight, Ioannis Souldatos. 2016-02-02. Hanf Number for Scott Sentences of Computable Structures. https://arxiv.org/abs/1602.01156

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