Turing Degrees of Isomorphism Types of Algebraic Objects

Details Turing Degrees of Isomorphism Types of Algebraic Objects Turing Degrees of Isomorphism Types of Algebraic Objects

2005-07-06

arxiv.org/abs/math/0507128v1

Mathematics

Logic

17 pages

Scientific paper

The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$. The Turing degree of the isomorphism type of $\mathcal{A}$, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsion-free abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees.

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