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Computable trees of Scott rank \(\omega_1^{CK}\), and computable approximation. (English) Zbl 1112.03039

In [“An example concerning Scott heights”, J. Symb. Log. 46, 301–318 (1981; Zbl 0501.03018)], M. Makkai produced an arithmetical structure of Scott rank \(\omega_{1}^{CK}\). Knight and Young later constructed computable structures of rank \(\omega_{1}^{CK}\). In this paper, the authors show that there are computable trees of Scott rank \(\omega_{1}^{CK}\). The paper also presents an example of a computable tree of rank \(\omega_{1}^{CK}\) which is strongly computably approximable. This is the first example of a structure of Scott rank \(\omega_{1}^{CK}\) for which strong computable approximability is known. This example is related to the following open problem: Whether all structures of noncomputable Scott rank are strongly computably approximable [S. S. Goncharov and J. F. Knight, “Computable structure and non-structure theorems”, Algebra Logika 41, No. 6, 639–681 (2002); translation in Algebra Logic 41, No. 6, 351–373 (2002; Zbl 1034.03044)].

MSC:

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