Ebook: Degree Spectra of Relations on a Cone
Author: Matthew Harrison-Trainor
- Tags: Unsolvability (Mathematical logic), Conic sections., Angles (Geometry)-Measurement.
- Series: Memoirs of the American Mathematical Society Ser.
- Year: 2018
- Publisher: American Mathematical Society
- City: Providence, United States
- Edition: 1
- Language: English
- pdf
Let $mathcal A$ be a mathematical structure with an additional relation $R$. The author is interested in the degree spectrum of $R$, either among computable copies of $mathcal A$ when $(mathcal A,R)$ is a "natural" structure, or (to make this rigorous) among copies of $(mathcal A,R)$ computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov--that, assuming an effectiveness condition on $mathcal A$ and $R$, if $R$ is not intrinsically computable, then its degree spectrum contains all c.e. degrees--the author shows that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e. degrees.
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