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

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

2008-03-22

arxiv.org/abs/0803.3296v1

Mathematics

Logic

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

Scientific paper

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.

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