Archive for Mathematical Logic

, Volume 43, Issue 4, pp 543–555 | Cite as

Effectiveness for infinite variable words and the Dual Ramsey Theorem

Article

Abstract.

We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k,l) and OVW(k,l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson [1] to show that this statement implies ACA0 over RCA0. We show that neither VW(2,2) nor OVW(2,2) is provable in WKL0. These results give partial answers to questions posed by Friedman and Simpson [3].

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References

  1. 1.
    Carlson, T.J., Simpson, S.G.: A dual form of Ramsey’s Theorem. Advances in Mathematics 53, 265–290 (1984)MathSciNetMATHGoogle Scholar
  2. 2.
    Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B.: Ed., ‘‘Handbook of Recursive Model Theory (Two Volumes)’’ Elsevier, Amsterdam 1998Google Scholar
  3. 3.
    Friedman, H., Simpson, S.G.: Issues and problems in reverse mathematics, in ‘‘Computability Theory and Its Applications: Current Trends and Open Problems’’ (P.A. Cholak, S. Lempp, M. Lerman & R.A. Shore, Ed.) American Mathematical Society, Providence 2000, 127–144Google Scholar
  4. 4.
    Jockusch, C.G.: Jr., Ramsey’s theorem and recursion theory. Journal of Symbolic Logic 37, 268–280 (1972)Google Scholar
  5. 5.
    Ramsey, F.P.: On a problem of formal logic. Proceedings of the London Mathematical Society 30, 264–286 (1930)MATHGoogle Scholar
  6. 6.
    Simpson, S.G.: Recursion theoretic aspects of the Dual Ramsey Theorem, in ‘‘Recursion Theory Week’’ (H.-D. Ebbinghaus, G.H. Müller & G.E. Sacks, Ed.) Springer- Verlag, New York 1985, pp. 356–371Google Scholar
  7. 7.
    Simpson, S.G.: ‘‘Subsystems of second order arithmetic’’, Springer-Verlag, New York 1999Google Scholar
  8. 8.
    Slaman, T.A.: A note on Dual Ramsey Theorem, 4 pages, unpublished, January 1997Google Scholar
  9. 9.
    Soare, R.I.: ‘‘Recursively enumerable sets and degrees’’, Springer-Verlag, New York 1987Google Scholar
  10. 10.
    Specker, E.: Ramsey’s theorem does not hold in recursive set theory, in ‘‘Logic Colloquium ‘69 (Proc. Summer School and Colloq., Manchester, 1969)’’, North-Holland, Amsterdam 1971, pp. 439–442Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsIndiana UniversityUSA
  2. 2.Department of MathematicsUniversity of Connecticut StorrsUSA

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