Notre Dame Journal of Formal Logic

Realizing Levels of the Hyperarithmetic Hierarchy as Degree Spectra of Relations on Computable Structures

Denis R. Hirschfeldt and Walker M. White

Abstract

We construct a class of relations on computable structures whose degree spectra form natural classes of degrees. Given any computable ordinal $\alpha$ and reducibility r stronger than or equal to m-reducibility, we show how to construct a structure with an intrinsically $\Sigma_\alpha$ invariant relation whose degree spectrum consists of all nontrivial $\Sigma_\alpha$ r-degrees. We extend this construction to show that $\Sigma_\alpha$ can be replaced by either $\Pi_\alpha$ or $\Delta_\alpha$.

Article information

Source
Notre Dame J. Formal Logic, Volume 43, Number 1 (2002), 51-64.

Dates
First available in Project Euclid: 15 December 2003

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1071505769

Digital Object Identifier
doi:10.1305/ndjfl/1071505769

Mathematical Reviews number (MathSciNet)
MR2033315

Zentralblatt MATH identifier
1048.03035

Subjects
Primary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]
Secondary: 03C15: Denumerable structures 03C57: Effective and recursion-theoretic model theory [See also 03D45] 03D55: Hierarchies 03D30: Other degrees and reducibilities

Keywords
degree spectra of relations computable structures computable model theory hyperarithmetic hierarchy

Citation

Hirschfeldt, Denis R.; White, Walker M. Realizing Levels of the Hyperarithmetic Hierarchy as Degree Spectra of Relations on Computable Structures. Notre Dame J. Formal Logic 43 (2002), no. 1, 51--64. doi:10.1305/ndjfl/1071505769. https://projecteuclid.org/euclid.ndjfl/1071505769


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