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arXiv · 2209.04524

Degrees of categoricity and treeable degrees

Barbara F. Csima·Dino Rossegger
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Abstract

We give a characterization of the strong degrees of categoricity of computable structures greater or equal to $\mathbf 0''$. They are precisely the \emph{treeable} degrees -- the least degrees of paths through computable trees -- that compute $\mathbf 0''$. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree $\mathbf d$ with $\mathbf 0^{(\alpha)}\leq \mathbf d\leq \mathbf 0^{(\alpha+1)}$ for $\alpha$ a computable ordinal greater than $2$ is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree $\mathbf d$ with $\mathbf 0'\leq \mathbf d\leq \mathbf 0''$ is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree $\mathbf d$ with $\mathbf 0'< \mathbf d< \mathbf 0''$ that is not the degree of categoricity of a rigid structure.

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Barbara F. Csima, Dino Rossegger. 2022-09-09. Degrees of categoricity and treeable degrees. https://arxiv.org/abs/2209.04524

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