2016/06/20 by Knight, Julia, Saraph, Vikram · 1 citation
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1606.06353
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 Σ3 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-Σ2" (the conjunction of a computable Σ2 sentence and a computable Π2 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 ℚ. We show that for some of these groups, the computable Σ3 Scott sentence is best possible, while for others, there is a computable d-Σ2 Scott sentence.