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- Publisher
- Springer Journals
- Copyright
- Copyright © 2017 by Springer-Verlag GmbH Germany
- Subject
- Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
- ISSN
- 0933-5846
- eISSN
- 1432-0665
- DOI
- 10.1007/s00153-017-0578-z
- Publisher site
- See Article on Publisher Site
Abstract
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 $$\varSigma _3$$ Σ 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- $$\varSigma _2$$ Σ 2 ” (the conjunction of a computable $$\varSigma _2$$ Σ 2 sentence and a computable $$\varPi _2$$ Π 2 sentence). In [9], this was 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 $$\mathbb {Q}$$ Q . We show that for some of these groups, the computable $$\varSigma _3$$ Σ 3 Scott sentence is best possible, while for others, there is a computable d- $$\varSigma _2$$ Σ 2 Scott sentence.
Journal
Archive for Mathematical Logic – Springer Journals
Published: Aug 19, 2017