Effective bounds from ineffective proofs in analysis: An application of functional interpretation and majorization

Abstract

AbstractWe show how to extract effective bounds Φ for ⋀u1⋀v ≤ytu⋁wηG0-sentences which depend on u only (i.e. ⋀u⋀v ≤y, tu⋁w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form(ϒ, δ, ρ, and τ are arbitrary finite types, η ≤ 2, G0 and F0 are quantifier-free, and s and t are closed terms). If τ ≤ 2, (*) can be weakened toThis is used to establish new conservation results about weak Konig's lemma. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers.