Metric fixed point theory and partial impredicativity

Abstract

We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in ${\mathsf{RCA}}_0$. 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 ${\mathsf{ATR}}_0$ and $Π^1_1\mbox{-}\mathsf{CA}_0$. We also exhibit several weakenings of Caristi's theorem that are equivalent to ${\mathsf{WKL}}_0$ and to ${\mathsf{ACA}}_0$.