## Notre Dame Journal of Formal Logic

### Ramsey's Theorem for Pairs and Provably Recursive Functions

#### Abstract

This paper addresses the strength of Ramsey's theorem for pairs ($RT^2_2$) over a weak base theory from the perspective of 'proof mining'. Let $RT^{2-}_2$ denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of $\Sigma^0_1$-induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). In the resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of $\forall\exists$-theorems.

There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles.

So from the perspective of proof unwinding the computational content of concrete proofs based on $RT^2_2$ the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of $RT^2_2$ that are primitive recursive in f.

#### Article information

Source
Notre Dame J. Formal Logic Volume 50, Number 4 (2009), 427-444.

Dates
First available: 11 February 2010

http://projecteuclid.org/euclid.ndjfl/1265899123

Digital Object Identifier
doi:10.1215/00294527-2009-019

Mathematical Reviews number (MathSciNet)
MR2598872

Zentralblatt MATH identifier
05778809

Subjects
Primary: 03F10 05D10 03F35

#### Citation

Kreuzer , Alexander; Kohlenbach , Ulrich. Ramsey's Theorem for Pairs and Provably Recursive Functions. Notre Dame Journal of Formal Logic 50 (2009), no. 4, 427--444. doi:10.1215/00294527-2009-019. http://projecteuclid.org/euclid.ndjfl/1265899123.

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