Computable trees of Scott rank ω 1CK, and computable approximation

Abstract

Makkai [10] produced an arithmetical structure of Scott rank ω₁^CK. In [9], Makkai’s example is made computable. Here we show that there are computable trees of Scott rank ω₁^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 ω₁^CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind.