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Intrinsic bounds on complexity and definability at limit levels. (English) Zbl 1201.03019
A structure is called computably categorical if every computable structure isomorphic to it is computably isomorphic to it. The structure is called relatively computably categorical iff this statement is true for all oracles. For example, a dense linear ordering without endpoints is relatively computably categorical. Long ago, Goncharov showed that a structure \({\mathcal A}\) is relatively computably categorical iff it has a c.e. Scott family. Further, there are structures that are computably categorical without such families and hence the notions differ. This material was extended to \(\Delta_\alpha^0\)-categoricity and \(\Sigma_\alpha^0\)-Scott families by Ash and Knight, where \(\alpha\) is a (notation for a) computable ordinal. S. Goncharov, V. Harizanov, J. Knight, C. McCoy, R. Miller and R. Solomon [Ann. Pure Appl. Logic 136, No. 3, 219–246 (2005; Zbl 1081.03033)] showed that for limit ordinals there exist \(\Delta_\alpha^0\)-categorical structures which are not relatively \(\Delta_\alpha^0\)-categorical, and the present paper completes the picture by showing this for \(\alpha\) a limit ordinal. The method uses codings of families, and an “\(\alpha\)-system argument”.

03C57 Computable structure theory, computable model theory
03C35 Categoricity and completeness of theories
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