Structures in Familiar Classes Which Have Scott Rank ${\omega }_1^{CK}$

Abstract

There are familiar examples of computable structures having various computable Scott ranks. There are also familiar structures, such as the Harrison ordering, which have Scott rank ${\omega }_1^{CK}+1$. Makkai produced a structure of Scott rank ${\omega }_1^{CK}$, which can be made computable, and simplified so that it is just a tree. In the present paper, we show that there are further computable structures of Scott rank ${\omega }_1^{CK}$ in the following classes: undirected graphs, fields of any characteristic, and linear orderings. The new examples share with the Harrison ordering, and the tree just mentioned, a strong approximability property.

Advances in Logic (Proceedings of the North Texas Logic Conference, October 8--10, 2004), Contemporary Mathematics 425 (2007), American Mathematical Society, 49--66