Metric fixed point theory and partial impredicativity

We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in [Formula: see text]. 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 [Formula: see text] and [Formula: see text]. We also exhibit several weakenings of Caristi's theorem that are equivalent to [Formula: see text] and to [Formula: see text]. This article is part of the theme issue 'Modern perspectives in Proof Theory'.