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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 that this upper bound is tight, i.e., for every $alpha < omega_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $alpha$ the class of Hausdorff rank $alpha$ linear orders is $pmb Sigma_{2alpha+2}$ complete.
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. For both kinds of groups, the computable ones all have computable $Sigma_3$ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are computable $d$-$Sigma_2$ (the conjunction of a computable $Sigma_2$ sentence and a computable $Pi_2$ sentence). This was already shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank $1$. These are exactly the additive subgroups of $mathbb{Q}$. We show that for some of these groups, the computable $Sigma_3$ Scott sentence is best possible, while for others, there is a computable $d$-$Sigma_2$ Scott sentence.
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 prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic.
We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a emph{single} $mathcal{L}_{omega_1,omega}$-sentence $psi$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $psi$ is consistently equal to $[aleph_0,aleph_{omega_1}]$ and also consistently equal to $[aleph_0,2^{aleph_1})$, where $2^{aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $psi$ is consistently equal to $[aleph_1,aleph_{omega_1}]$ and $[aleph_1,2^{aleph_1})$, where again $2^{aleph_1}$ is weakly inaccessible. This is the first example of an $mathcal{L}_{omega_1,omega}$-sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in [18]. (3) Consistently, $psi$ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in [1] and [2] of sentences with maximal models in countably many cardinalities. (4) $2^{aleph_0}<aleph_{omega_1}<2^{aleph_1}$ and there exists an $mathcal{L}_{omega_1,omega}$-sentence with models in $aleph_{omega_1}$, but no models in $2^{aleph_1}$. This relates to a conjecture by Shelah that if $aleph_{omega_1}<2^{aleph_0}$, then any $mathcal{L}_{omega_1,omega}$-sentence with a model of size $aleph_{omega_1}$ also has a model of size $2^{aleph_0}$. Our result proves that $2^{aleph_0}$ can not be replaced by $2^{aleph_1}$, even if $2^{aleph_0}<aleph_{omega_1}$.
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