Metric fixed point theory and partial impredicativity.
PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023;
381:20220012. [PMID:
37031705 DOI:
10.1098/rsta.2022.0012]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2022] [Accepted: 02/09/2023] [Indexed: 06/19/2023]
Abstract
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'.
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