The proof-theoretic strength of Ramsey's theorem for pairs and two colors

Ludovic Patey 1, 2 Keita Yokoyama 3
2 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : Ramsey's theorem for n-tuples and k-colors (RT n k) asserts that every k-coloring of [N] n admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two colors, namely, the set of its Π 0 1 consequences, and show that RT 2 2 is Π 0 3 conservative over IΣ 0 1. This strengthens the proof of Chong, Slaman and Yang that RT 2 2 does not imply IΣ 0 2 , and shows that RT 2 2 is finitistically reducible, in the sense of Simpson's partial realization of Hilbert's Program. Moreover, we develop general tools to simplify the proofs of Π 0 3-conservation theorems.
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Ludovic Patey, Keita Yokoyama. The proof-theoretic strength of Ramsey's theorem for pairs and two colors. Advances in Mathematics, Elsevier, 2018, 330, pp.1034 - 1070. ⟨10.1016/j.aim.2018.03.035⟩. ⟨hal-01888655⟩

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