Scott sentences for equivalence structures

  • Sara B. QuinnEmail author


For a computable structure \({\mathcal {A}}\), if there is a computable infinitary Scott sentence, then the complexity of this sentence gives an upper bound for the complexity of the index set \(I({\mathcal {A}})\). If we can also show that \(I({\mathcal {A}})\) is m-complete at that level, then there is a correspondence between the complexity of the index set and the complexity of a Scott sentence for the structure. There are results (Calvert et al. in Algebra Logic 45:306–325, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012) that suggest that these complexities will always match. However, it was shown in Knight and McCoy (Arch Math Logic 53:519–524, 2014) that there is a structure (a particular subgroup of \({\mathbb {Q}}\)) for which the index set is m-complete \(d-\varSigma ^0_2\), though there is no computable \(d-\varSigma _2\) Scott sentence. In the present paper, we give an example of a particular equivalence structure for which the index set is m-complete \(\varPi _3^0\) but for which there is no computable \(\varPi _3\) Scott sentence. There is, however, a computable \(\varPi _3\) pseudo-Scott sentence for the structure, that is, a sentence that acts as a Scott sentence if we only consider computable structures.


Index sets Scott sentences Equivalence structures 

Mathematics Subject Classification

03D45 03C57 



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Authors and Affiliations

  1. 1.Dominican UniversityRiver ForestUSA

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