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 $ACA_0$, the comprehension scheme for arithmetical formulas, within the base theory $RCA_0$, 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 $RCA_0$. In particular, Ramsey's Theorem for Pairs is not conservative over $RCA_0$ for $\Pi^0_4$-sentences.