Structures in Familiar Classes Which Have Scott Rank $ω_1^{CK}$

Calvert, Wesley; Goncharov, Sergey S.; Knight, Julia F.

Structures in Familiar Classes Which Have Scott Rank $ω_1^{CK}$

Authors

Calvert, Wesley

Goncharov, Sergey S.

Knight, Julia F.

Description

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 $ω_1^{CK}+1$. Makkai produced a structure of Scott rank $ω_1^{CK}$, 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 $ω_1^{CK}$ 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