Gödel functional interpretation and weak compactness

Abstract

In recent years, proof theoretic transformations (so-called proof interpretations) that are based on extensions of monotone forms of Gödel’s famous functional (‘Dialectica’) interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for bounded closed and convex subsets of an abstract (not necessarily separable) Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω∗ definable from bar recursion (in the sense of Spector) of lowest type. While a case study on the analysis of strong convergence results (due to Browder and Wittmann resp.) that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an explicit bound on a metastable (in the sense of T. Tao) version of this theorem that is primitive recursive relative to (a somewhat restricted form of) Ω∗.In the current paper we for the first time give the construction of Ω∗ (in the form needed for the unwinding of Baillon’s theorem). As a corollary to the fine analysis of the use of bar recursion in this construction we obtain that Ω∗ elevates arguments in Tn at most to resulting functionals in Tn+2 (here Tn is the fragment of Gödel’s T with primitive recursion restricted to the type level n). In particular, one can conclude from this that our bound on Baillon’s theorem is at least definable in T4.