Hanf number for Scott sentences of computable structures

Archive for Mathematical Logic 57 (7-8):889-907 (2018)
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The Hanf number for a set S of sentences in \ is the least infinite cardinal \ such that for all \, if \ has models in all infinite cardinalities less than \, then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \. The same argument proves that \ is the Hanf number for Scott sentences of hyperarithmetical structures.



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Model theory for infinitary logic.H. Jerome Keisler - 1971 - Amsterdam,: North-Holland Pub. Co..
Computable structures and the hyperarithmetical hierarchy.C. J. Ash - 2000 - New York: Elsevier. Edited by J. Knight.
Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.

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