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

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


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ash, C.J. [1972], Some problems in Mathematical Logic. D. Phil. Thesis, Oxford University.Google Scholar
  2. Dushnik, B. and E.W. Miller [1940], Concerning similarity transformations of linearly ordered sets. Bull Amer. Math. Soc, 49, 322–326.MathSciNetCrossRefGoogle Scholar
  3. Goncharov, S.S. [1975], Some properties of the constructivization of Boolean algebras. Sibirski Math. Zh, 16, 203–214.zbMATHCrossRefGoogle Scholar
  4. Kierstead, H. [1987], On Π-automorphisms of recursive linear orderings. J. Symbolic Logic, 52, 681–688.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Langford, C.H. [1927], Theorems on deducibility. Annals of Mathematics, 28, 459–471.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Moses, M. [1983], Recursive properties of isomorphism types. J. Aust. Math. Soc. Series A, 34, 269–286.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Moses, M. [1984], Recursive linear orders with recursive successivities. Annals of Pure & Appl. Logic, 27, 253–264.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Moses, M., Quantifier elimination in linear orders. In preparation.Google Scholar
  9. Remmel, J.B. [1981], Recursively categorical linear orderings. Proc. Amer. Math. Soc, 83, 379–386.MathSciNetzbMATHGoogle Scholar
  10. Rosenstein, J.G. [1982], Linear Orderings. Academic Press.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

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

Personalised recommendations