Structures in Familiar Classes Which Have Scott Rank $\omega_1^{CK}$

TL;DR

This paper constructs new computable structures with Scott rank exactly ^{CK} in classes like graphs, fields, and linear orders, expanding the known examples and highlighting their approximability properties.

Contribution

It introduces new computable structures of Scott rank ^{CK} in various classes, demonstrating their strong approximability and broadening the understanding of Scott ranks.

Findings

Existence of computable structures with Scott rank ^{CK} in graphs, fields, and linear orders.

These structures share approximability properties with known examples like the Harrison ordering.

The results extend the landscape of structures with high Scott rank in computability theory.

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.