Computable Trees of Scott Rank $\omega _{1}^{\mathit{CK}}$ , and Computable Approximation

Journal of Symbolic Logic 71 (1):283 - 298 (2006)
Makkai [10] produced an arithmetical structure of Scott rank $\omega _{1}^{\mathit{CK}}$. In [9]. Makkai's example is made computable. Here we show that there are computable trees of Scott rank $\omega _{1}^{\mathit{CK}}$. We introduce a notion of "rank homogeneity". In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated "group trees" of [10] and [9]. Using the same kind of trees, we obtain one of rank $\omega _{1}^{\mathit{CK}}$ that is "strongly computably approximable". We also develop some technology that may yield further results of this kind.
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