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n-Recursive Linear Orders without (n+1)-Recursive Copies

  • Michael Moses
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)

Abstract

An algebraic structure, with universe ω, is said to be recursive if, and only if, the quantifier-free formulae in its language uniformly denote recursive relations in the structure. This concept generalizes to that of an n-recursive structure, in which the Σn formulae uniformly denote recursive relations. I present a sequence of simply defined relations in the language of linear order and show that a recursive linear order of type ωk (with k < ω) is n-recursive if, and only if, it has the first n relations in the sequence recursive. I can then show that, for each k, and each n ≤ (2k − 2), ωk has an n-recursive copy that is not (n+1)-recursive. This means that, for each n, there is an n-recursive but not (n+1)-recursive linear order (of some type ωk). I show, in fact, that there is, for each n, a linear order (of some type ωk · τ) that is n-recursive but has no (n+1)-recursive copy.

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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Michael Moses
    • 1
  1. 1.Department of MathematicsThe George Washington UniversityUSA

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