Skip to main content
    • Aa
    • Aa

On the strength of Ramsey's theorem for pairs

  • Peter A. Cholak (a1), Carl G. Jockusch (a2) and Theodore A. Slaman (a3)

We study the proof–theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RTkn denote Ramsey's theorem for k–colorings of n–element sets, and let RT<∞n denote (∀k)RTkn. Our main result on computability is: For any n ≥ 2 and any computable (recursive) k–coloring of the n–element sets of natural numbers, there is an infinite homogeneous set X with X″ ≤T 0(n). Let IΣn and BΣn denote the Σn induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models of arithmetic enables us to show that RCA0 + IΣ2 + RT22 is conservative over RCA0 + IΣ2 for Π11 statements and that RCA0 + IΣ3 + RT<∞2 is Π11-conservative over RCA0 + IΣ3. It follows that RCA0 + RT22 does not imply BΣ3. In contrast, J. Hirst showed that RCA0 + RT<∞2 does imply BΣ3, and we include a proof of a slightly strengthened version of this result. It follows that RT<∞2 is strictly stronger than RT22 over RCA0.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Jeremy Avigad , [1996], Formalizing forcing arguments in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 82, no. 2, pp. 165191.

Matt Fairtlough and Stanley S. Wainer [1998], Hierarchies of provably recursive functions, Handbook of proof theory, Studies in Logic and the Foundations of Mathematics, vol. 137, North-Holland, Amsterdam.

Ronald L. Graham , Bruce L. Rothschild , and Joel H. Spencer [1990], Ramsey theory, second ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, A Wiley-Interscience Publication.

Petr Hájek and Pavel Pudlák , [1993], Metamathematics of first-order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin.

Carl G. Jockusch Jr. [1968], Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131, pp. 420436.

Carl G. Jockusch Jr. [1973], Upward closure and cohesive degrees, Israel Journal of Mathematics, vol. 15, pp. 332335.

Carl G. Jockusch Jr., and Robert I. Soare , [1972], Π10 classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173, pp. 3356.

Carl G. Jockusch Jr. and Frank Stephan [1993], A cohesive set which is not high, Mathematical Logic Quarterly, vol. 39, pp. 515530.

Carl G. Jockusch Jr. and Frank Stephan [1997], Correction to “A cohesive set which isnothigh”, Mathematical Logic Quarterly, vol. 43, p. 569.

G. E. Mints [1973], Quantifer-free and one quantifier systems, Journal of Soviet Mathematics, vol. 1, pp. 7184.

J. B. Paris , [1980], A hierarchy of cuts in models of arithmetic, Model theory of algebra and arithmetic (Proceedigs, Karpacz, Poland 1979), Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, New York, pp. 312337.

Charles Parsons , [1970], On a number theoretic choice schema and its relation to induction, Intuitionism and proof theory (Proceedings of a Conference, Buffalo, N.Y., 1968), North-Holland, Amsterdam, pp. 459473.

Dana Scott , [1962], Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Providence, R.I.), Proceedings of Symposia in Pure Mathematics, no. 5, American Mathematical Society, pp. 117121.

Robert I. Soare , [1987], Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Heidelberg.

C. Spector , [1956], On the degrees of recursive unsolvability, Annals of Mathematics (2), vol. 64, pp. 581592.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *