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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 $varphiin S$, 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.
We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. Fo
Given a countable scattered linear order $L$ of Hausdorff rank $alpha < omega_1$ we show that it has a $dtext{-}Sigma_{2alpha+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain th
In [13] the authors show that if $mu$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<mu$, and $K$ satisfies joint embedding (amalgamation) cofinally below $mu$, then $K$ satisfies joint embedding (amalgamation)
We obtain a computable structure of Scott rank omega_1^{CK} (call this ock), and give a general coding procedure that transforms any hyperarithmetical structure A into a computable structure A such that the rank of A is ock, ock+1, or < ock iff the same is true of A.