Effective Bounds from Ineffective Proofs in Analysis: An Application of Functional Interpretation and Majorization

We show how to extract effective bounds Φ for ∀u∀v ≤γ tu∃wG0–sentences which depend on u only (i.e. ∀u∀v ≤γ tu∃w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form (∗)∀x∃y ≤ρ sx∀zF0 (δ, ρ, τ are arbitrary finite types, η ≤ 2, G0, F0 are quantifier–free and s, t closed terms). If τ ≤ 2, (∗) can be weakened to ∀x, z∃y ≤ρ sx∀z̃ ≤τ zF0. This is used to establish new conservation results about weak Knig’s lemma WKL. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers.