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On the strength of Ramsey's theorem for pairs

Published online by Cambridge University Press:  12 March 2014

Peter A. Cholak ,
Carl G. Jockusch  and
Theodore A. Slaman
Peter A. Cholak
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, USA, E-mail: Peter.Cholak.l@nd.edu
Carl G. Jockusch
Affiliation:
Department of Mathematics, University of Illinois, at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801-2975, USA, E-mail: jockusch@math.uiuc.edu
Theodore A. Slaman
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA, E-mail: slaman@math.berkeley.edu
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Abstract

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.

Keywords

Ramsey's Theoremconservationreverse mathematicsrecursion theorycomputability theory
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 1 - 55
Copyright
Copyright © Association for Symbolic Logic 2001

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