Index sets and Scott sentences

Abstract

For a computable structure ${\mathcal{A}}$, there may not be a computable infinitary Scott sentence. When there is a computable infinitary Scott sentence ${\varphi}$, then the complexity of the index set ${I(\mathcal{A})}$ is bounded by that of ${\varphi}$. There are results (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print) giving “optimal” Scott sentences for structures of various familiar kinds. These results have been driven by the thesis that the complexity of the index set should match that of an optimal Scott sentence (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012). In this note, it is shown that the thesis does not always hold. For a certain subgroup of ${\mathbb{Q}}$, there is no computable d-${\Sigma_2}$ Scott sentence, even though (as shown in Ash and Knight in Scott sentences for certain groups, pre-print) the index set is d-${\Sigma^0_2}$.