## 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

### Keywords:

Scott rank; computable trees; computable approximation

### Citations:

Zbl 0501.03018; Zbl 1034.03044
Full Text:

### References:

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