Elimination of Skolem functions for monotone formulas in analysis

Abstract

In this paper a new method, elimination of Skolem functions for monotone formulas, is developed which makes it possible to determine precisely the arithmetical strength of instances of various non-constructive function existence principles. This is achieved by reducing the use of such instances in a given proof to instances of certain arithmetical principles. Our framework are systems \({\cal T}^{\omega} :={\rm G}_n{\rm A}^{\omega} +{\rm AC}\)-qf \(+\Delta\), where (G\(_n\)A\(^{\omega})_{n \in {\Bbb N}}\) is a hierarchy of (weak) subsystems of arithmetic in all finite types (introduced in [14]), AC-qf is the schema of quantifier-free choice in all types and \(\Delta\) is a set of certain analytical principles which e.g. includes the binary Konig's lemma. We apply this method to show that the arithmetical closures of single instances of \(\Pi^0_1\)-comprehension and -choice contribute to the growth of extractable bounds from proofs relative to \({\cal T}^\omega\) only by a primitive recursive functional in the sense of Kleene. In subsequent papers these results are widely generalized and the method is used to determine the arithmetical content of single sequences of instances of the Bolzano-Weierstras principle for bounded sequences in \({\Bbb R}^d\), the Ascoli-lemma and others.