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 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 ∀∃-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 .

Term extraction and Ramsey's theorem for pairs

“…As a consequence it also gives a proof theoretic account of conservation results for those principles. This paper extends our previous treatment of Ramsey's theorem for pairs in [34], where only single instances of Ramsey's theorem are discussed, to the full second order closure of those principles.…”

RAMSEY’S THEOREM FOR PAIRS ANDKCOLORS AS A SUB-CLASSICAL PRINCIPLE OF ARITHMETIC

“…We modify Erdős-Rado's proof of RT 2 k (see e.g. [10]) to obtain a proof of 3-LLPO =⇒ RT 2 k (Σ 0 0 ) over HA. It is enough to prove that if {c a | a ∈ N} is a recursive family of recursive colorings, a finite number of statements in 3-LLPO imply that there are predicates C 0 (., c), .…”

“…There is some Erdős' tree recursively enumerable in the coloring (e.g. [10,3]). Assume there exists some infinite k-ary Erdős' tree V .…”

“…It is not too difficult to take the proof by Erdős and Rado of Ramsey's Theorem for pairs (e.g. [10,9]) and to formalize it in HA extended by some classical principle. In this way we could prove that, for every natural number k ≥ 2, 4-LLPO (i.e.…”

An intuitionistic version of Ramsey's Theorem and its use in Program Termination

“…Bellin in [4] applied the no-counterexample interpretation to Ramsey's Theorem, while Oliva and Powell in [22], by starting from a formalization of Erdős and Rado's proof of Ramsey's Theorem given by Kreuzer and Kohlenbach in [18], used the Dialectica interpretation. They approximated the homogeneous set by a set which may stand any test checking whether some initial segment is homogeneous.…”