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Index sets for classes of high rank structures. (English) Zbl 1145.03021
The authors prove that the index set of the class of all structures of non-computable Scott rank is \(\Sigma_1^1\)-complete; the index set of the class of structures of Scott rank \(\omega^{CK}_1\) is \(\Pi_2^0\)-complete relative to Klenee’s \({\mathcal O}\); and the index set of the class of structures of Scott rank \(\omega^{CK}_1+1\) is \(\Sigma_1^0\)-complete relative to \({\mathcal O}\).

MSC:
03D45 Theory of numerations, effectively presented structures
03C57 Computable structure theory, computable model theory
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