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'.

Computability theory as a philosophical achievement

Artificial intelligence plays an important role in contemporary medicine. In this short note, we emphasize that philosophy played a role in the development of artificial intelligence. We argue that research in computability theory, the theoretical foundation of modern computer science, was motivated by a philosophical question: can we characterize precisely the class of problems that can be solved by algorithms? We suggest that reflecting on the connection between philosophy and artificial intelligence helps realize that philosophical and scientific progress are connected.