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.

中文翻译：

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等价结构的斯科特句子

对于可计算结构\（{\ mathcal {A}} \），如果存在可计算的非限定性Scott句子，则该句子的复杂度为索引集\（I（{\ mathcal { A}}）\）。如果我们还可以证明\（I（{\ mathcal {A}}）\）是m-complete在该级别上，则索引集的复杂度与该结构的Scott句子的复杂度之间存在对应关系。有结果（Calvert等人，Algebra Logic 45：306-325，2006； Carson等人，Trans Am Math Soc 364：5715-5728，2012； Knight和Saraph在某些组的Scott句子中，预印； McCoy和Wallbaum在Trans Am Math Soc 364：5729–5734，2012）中提出，这些复杂性将永远匹配。但是，在Knight和McCoy（2014年Arch Math Logic 53：519–524）中显示，存在一个结构（\（{\ mathbb {Q}} \}的特定子组），其索引集为m -complete \（d- \ varSigma ^ 0_2 \），尽管没有可计算的\（d- \ varSigma _2 \）斯科特一句话。在本论文中，我们给出一个特定等价结构的量，索引集合是一个例子米-complete \（\ varPi _3 ^ 0 \） ，但对于其中不存在可计算\（\ varPi _3 \）斯科特句子。但是，该结构有一个可计算的\（\ varPi _3 \）伪Scott句子，即，如果仅考虑可计算的结构，则该句子将充当Scott句子。