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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
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
We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic.
Classical scope-assignment strategies for multi-quantifier sentences involve quantifier phrase (QP)-movement. More recent continuation-based approaches provide a compelling alternative, for they interpret QPs in situ - without resorting to Logical Fo