Metric fixed point theory and partial impredicativity

Abstract

We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in R C A 0 . Furthermore, we show that Caristi’s fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between A T R 0 and Π 1 1 - C A 0 . We also exhibit several weakenings of Caristi’s theorem that are equivalent to W K L 0 and to A C A 0 . This article is part of the theme issue ‘Modern perspectives in Proof Theory’.