Journal of Symbolic Logic

On the Strength of Ramsey's Theorem for Pairs

Peter A. Cholak, Carl G. Jockusch, and Theodore A. Slaman

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We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT$^n_k$ denote Ramsey's theorem for k-colorings of n-element sets, and let RT$^n_{<\infty}$ denote ($\forall k)RT^n_k$. Our main result on computability is: For any n $\geq$ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' $\leq_T 0^{(n)}$. Let $I\Sigma_n$ and $B\Sigma_n$ denote the $\Sigma_n$ induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low$_2$) to models of arithmetic enables us to show that $RCA_0 + I\Sigma_2 + RT^2_2$ is conservative over $RCA_0 + I\Sigma_2$ for $\Pi^1_1$ statements and that $RCA_0 + I\Sigma_3 + RT^2_{<\infty}$, is $\Pi^1_1$-conservative over $RCA_0 + I\Sigma_3$. It follows that $RCA_0 + RT^2_2$ does not imply $B\Sigma_3$. In contrast, J. Hirst showed that $RCA_0 + RT^2_{<\infty}$ does imply $B\Sigma_3$, and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{<\infty}$ is strictly stronger than $RT^2_2$ over $RCA_0$.

Article information

J. Symbolic Logic, Volume 66, Issue 1 (2001), 1-55.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]
Secondary: 03C62: Models of arithmetic and set theory [See also 03Hxx] 03D30: Other degrees and reducibilities 03D80: Applications of computability and recursion theory

Ramsey's Theorem Conservation Reverse Mathematics Recursion Theory Computability Theory


Cholak, Peter A.; Jockusch, Carl G.; Slaman, Theodore A. On the Strength of Ramsey's Theorem for Pairs. J. Symbolic Logic 66 (2001), no. 1, 1--55.

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