The prooftheoretic strength of Ramsey's theorem for pairs and two colors
Abstract
Ramsey's theorem for $n$tuples and $k$colors ($\mathsf{RT}^n_k$) asserts that every kcoloring of $[\mathbb{N}]^n$ admits an infinite monochromatic subset. We study the prooftheoretic strength of Ramsey's theorem for pairs and two colors, namely, the set of its $\Pi^0_1$ consequences, and show that $\mathsf{RT}^2_2$ is $\Pi^0_3$ conservative over $\mathsf{I}\Sigma^0_1$. This strengthens the proof of Chong, Slaman and Yang that $\mathsf{RT}^2_2$ does not imply $\mathsf{I}\Sigma^0_2$, and shows that $\mathsf{RT}^2_2$ 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 $\Pi^0_3$conservation theorems.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1601.00050
 Bibcode:
 2016arXiv160100050P
 Keywords:

 Mathematics  Logic
 EPrint:
 32 pages