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Scott sentences for certain groups

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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 \(\varSigma _3\) 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-\(\varSigma _2\)” (the conjunction of a computable \(\varSigma _2\) sentence and a computable \(\varPi _2\) 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 \(\mathbb {Q}\). We show that for some of these groups, the computable \(\varSigma _3\) Scott sentence is best possible, while for others, there is a computable d-\(\varSigma _2\) Scott sentence.

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Authors and Affiliations

  1. University of Notre Dame, Notre Dame, IN, USA

    Julia F. Knight

  2. Brown University, Providence, RI, USA

    Vikram Saraph

Authors
  1. Julia F. Knight
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  2. Vikram Saraph
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Correspondence to Vikram Saraph.

Additional information

Since this paper was written (in 2013), there have been further results obtained. Ho [15] showed that the computable finitely generated groups in several further classes also have computable d-\(\varSigma _2\) Scott sentences. He also showed that for a finitely generated computable group G, if there is a computable \(\varSigma _2\) formula that defines a non-empty set of generating tuples, then there is a computable d-\(\varSigma _2\) Scott sentence. Still more recently, Harrison-Trainor and Ho [14] characterized the finitely generated groups that have a d-\(\varSigma _2\) Scott sentence as those that do not have a generating tuple \(\bar{a}\) and a further tuple \(\bar{b}\), not a generating tuple, such that all existential formulas true of \(\bar{b}\) are true of \(\bar{a}\). They gave an example of a computable finitely generated group that does not have a d-\(\varSigma _2\) Scott sentence. Alvir et al. [1] gave a different characterization. For a finitely generated group G, there is a d-\(\varSigma _2\) Scott sentence if and only if for every (some) generating tuple \(\bar{a}\), the orbit of \(\bar{a}\) is defined by a \(\varPi _1\) formula. For a computable finitely generated group, there is a computable d-\(\varSigma _2\) Scott sentence if and only if for every (some) generating tuple \(\bar{a}\), the orbit of \(\bar{a}\) is defined by a computable \(\varPi _1\) formula.

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Knight, J.F., Saraph, V. Scott sentences for certain groups. Arch. Math. Logic 57, 453–472 (2018). https://doi.org/10.1007/s00153-017-0578-z

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