On the strength of Ramsey’s theorem

Abstract

We show that, for every partition F of the pairs of natural numbers and for every set C, if C is not recursive in F then there is an infinite set H, such that H is homogeneous for F and C is not recursive in H. We conclude that the formal statement of Ramsey’s Theorem for Pairs is not strong enough to prove ACA0, the comprehension scheme for arithmetical formulas, within the base theory RCA0, the comprehension scheme for recursive formulas. We also show that Ramsey’s Theorem for Pairs is strong enough to prove some sentences in first order arithmetic which are not provable within RCA0. In particular, Ramsey’s Theorem for Pairs is not conservative over ACA0 for П04-sentences. © 1995, Duke University Press. All Rights Reserved.