Metric fixed point theory and partial impredicativity.

Area of Science:

• Mathematical analysis
• Set theory
• Foundations of mathematics

Background:

• Fixed-point theorems are fundamental in various mathematical fields.
• Understanding the axiomatic strength of these theorems is crucial for mathematical logic and proof theory.
• The relationship between different fixed-point theorems and foundational principles remains an active area of research.

Purpose of the Study:

• To determine the provability of the Priess-Crampe & Ribenboim fixed point theorem within a specific axiomatic system.
• To establish the precise logical relationship between Caristi's fixed point theorem and foundational principles.
• To explore weakenings of Caristi's theorem and their equivalences to specific axioms.

Main Methods:

• Axiomatic proof-theoretic analysis.
• Logical equivalences and strict inequalities between mathematical statements.
• Investigating fixed-point theorems for Baire and Borel functions.

Main Results:

• The Priess-Crampe & Ribenboim fixed point theorem is shown to be provable in [Formula: see text].
• Caristi's fixed point theorem for Baire and Borel functions is proven equivalent to the transfinite leftmost path principle.
• This principle is shown to lie strictly between [Formula: see text] and [Formula: see text].
• Several weakenings of Caristi's theorem are identified and shown to be equivalent to [Formula: see text] and [Formula: see text].

Conclusions:

• The study precisely positions the axiomatic strength of key fixed-point theorems within the mathematical landscape.
• It clarifies the logical dependencies between these theorems and foundational principles like the transfinite leftmost path principle.
• The findings contribute to a deeper understanding of proof theory and the structure of mathematical reasoning.