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arXiv · 0803.3296

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

Wesley Calvert·Sergey S. Goncharov·Julia F. Knight
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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 $ω_1^{CK}+1$. 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.

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Wesley Calvert, Sergey S. Goncharov, Julia F. Knight. 2008-03-22. Structures in Familiar Classes Which Have Scott Rank $ω_1^{CK}$. https://arxiv.org/abs/0803.3296

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