Ultrametric fixed points in reduced axiomatic systems

TL;DR

This paper demonstrates that a fixed point theorem for ultrametric spaces can be proven within a weaker axiomatic system using the Brezis-Browder ordering principle, reducing the reliance on stronger set-theoretic assumptions.

Contribution

It provides a new proof of a fixed point result in ultrametric spaces within a reduced axiomatic framework, extending the understanding of foundational requirements.

Findings

Fixed point theorem proven in ZF-AC+DC system

Utilizes Brezis-Browder ordering principle for the proof

Shows the result holds without full Axiom of Choice

Abstract

The Brezis-Browder ordering principle [Advances Math., 21 (1976), 355-364] is used to get a proof, in the reduced axiomatic system (ZF-AC+DC), of a fixed point result [in the complete axiomatic system (ZF)] over Cantor complete ultrametric spaces due to Petalas and Vidalis [Proc. Amer. Math. Soc., 118 (1993), 819-821].