Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 43, Number 1 (2002), 51-64.
Realizing Levels of the Hyperarithmetic Hierarchy as Degree Spectra of Relations on Computable Structures
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 $\Delta_\alpha$.
Notre Dame J. Formal Logic, Volume 43, Number 1 (2002), 51-64.
First available in Project Euclid: 15 December 2003
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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