Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals

This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA )n∈IN of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA ω by quantifier–free choice AC–qf and analytical axioms Γ having the form ∀x∃y ≤ρ sx∀z F0 (including also a ‘non– standard’ axiom F which does not hold in the full set–theoretic model but in the strongly majorizable functionals): From a proof GnA +AC–qf + Γ ⊢ ∀u, k∀v ≤τ tuk∃w A0(u, k, v, w) one can extract a uniform bound Φ such that ∀u, k∀v ≤τ tuk∃w ≤ ΦukA0(u, k, v, w) holds in the full set–theoretic type structure. In case n = 2 (resp. n = 3) Φuk is a polynomial (resp. an elementary recursive function) in k, u := λx.max(u0, . . . , ux). In the present paper we show that for n ≥ 2, GnA +AC–qf+F proves a generalization of the binary Konig’s lemma yielding new conservation results since the conclusion of the above rule can be verified in Gmax(3,n)A ω in this case. In a subsequent paper we will show that many important ineffective analytical principles and theorems can be proved already in G2A +AC–qf+Γ for suitable Γ.