Ordinal analysis and the infinite Ramsey theorem

Abstract

The infinite Ramsey theorem is known to be equivalent to the statement 'for every set X and natural number n, the n-th Turing jump of X exists', over RCA 0 due to results of Jockusch [5]. By subjecting the theory RCA 0 augmented by the latter statement to an ordinal analysis, we give a direct proof of the fact that the infinite Ramsey theorem has proof-theoretic strength ε ω. The upper bound is obtained by means of cut elimination and the lower bound by extending the standard well-ordering proofs for ACA 0. There is a proof of this result due to McAloon [6], using model-theoretic and combinatorial techniques. According to [6], another proof appeared in an unpublished paper by Jäger. © 2012 Springer-Verlag.