## On the strength of Ramsey’s Theorem for pairs (2001)

Venue: | Journal of Symbolic Logic |

Citations: | 42 - 9 self |

### BibTeX

@ARTICLE{Cholak01onthe,

author = {Peter A. Cholak and Carl G. Jockusch and Theodore and A. Slaman},

title = {On the strength of Ramsey’s Theorem for pairs},

journal = {Journal of Symbolic Logic},

year = {2001},

pages = {1--55}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We study the proof–theoretic strength and effective content denote Ram-of the infinite form of Ramsey’s theorem for pairs. Let RT n k sey’s theorem for k–colorings of n–element sets, and let RT n < ∞ denote (∀k)RTn k. 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 I�n and B�n denote the �n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models is conservative of arithmetic enables us to show that RCA0 + I �2 + RT2 2 over RCA0 + I �2 for �1 1 statements and that RCA0 + I �3 + RT2 < ∞ is �1 1-conservative over RCA0 + I �3. It follows that RCA0 + RT2 2 does not imply B �3. In contrast, J. Hirst showed that RCA0 + RT2 < ∞ does imply B �3, and we include a proof of a slightly strengthened version of this result. It follows that RT2 < ∞ is strictly stronger than RT2 2 over RC A0. 1.

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Citation Context ... and we include a proof of a slightly strengthened version of this result. It follows that RT2 <∞ is strictly stronger than RT2 2 over RC A0. 1. Introduction Ramsey’s theorem was discovered by Ramsey =-=[24]-=- and used by him to solve a decision problem in logic. Subsequently it has been an important tool in logic and combinatorics. Definition 1.1. (i) [X] n = {Y ⊆ X : |Y | = n}. (ii) A k–coloring C of [X]... |

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Citation Context ...of RCA0. Hájek [6] did it first, by describing a recursion-theoretic model of WKL066 P. CHOLAK, C. JOCKUSCH AND T. SLAMAN internal to RCA0 (in fact, in I �1). Without knowing about this work, Avigad =-=[1]-=- did it by internalizing the forcing argument. Hájek [6] explicitly mentions that the argument extends to weak König’s lemma over I �n more generally. The key to this extension is that the truth of a ... |

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Citation Context ...ply an effectivization of the standard proof of Ramsey’s Theorem. Instead, the first step is to restrict the given computable coloring to a low2 r–cohesive set A, which exists by Jockusch and Stephan =-=[16]-=-, Theorem 2.5. Since for any a the color of the pair {a, b} is independent of b for sufficiently large b ∈ A, the coloring induces a coloring of [A] 1 which is � 0,A 2 . Then the relativization to A o... |

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Citation Context ...ormal statement in second order arithmetic. There has been much work done along these lines. For example, consider the independent work by Jockusch [13], Seetapun, and Slaman (see Seetapun and Slaman =-=[26]-=-). Our task in this paper is to review briefly the work that has been done and further this analysis. Before getting into details we mention two themes in this work that we would like to make explicit... |

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Citation Context ...in RCA0 are exactly the primitive recursive functions. This characterizes the �0 2 sentences provable from RCA0. (Fairtlough and Wainer [2] credits this result to Parsons [22], Mints [19] and Takeuti =-=[32]-=-.) Hence to get a negative answer (to the above question) one must show using RT 2 2 that some computable but not primitive recursive function (such as the Ackermann function) is provably total. The f... |

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