Term extraction and Ramsey's theorem for pairs

Abstract

In this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH).

Our main result on COH is that the type 2 functionals provably recursive from RCA₀ + COH + Π⁰₁-CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that WKL₀+Π⁰₁-CP+COH is Π⁰₂-conservative over PRA.

Recent work of the first author showed that Π⁰₁-CP + COH is equivalent to a weak variant of the Bolzano—Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs.

For Ramsey's theorem for pairs and two colors (RT²₂) we obtain the upper bounded that the type 2 functionals provable recursive relative to RCA₀+Σ⁰₂-IA + RT²₂ are in T₁. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T₁-bounds from proofs that use RT²₂. Moreover, this yields a new proof of the fact that WKL₀+Σ⁰₂-IA + RT²₂ is Π⁰₃-conservative over RCA₀+Σ⁰₂-IA.

The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using Π⁰₁-comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for RT²₂).