On the relation of three theorems of analysis to the axiom of choice

TL;DR

This paper explores the logical strength of certain theorems in analysis by linking them to the axiom of choice, and investigates the axiomatic requirements for the uniform boundedness principle.

Contribution

It establishes the equivalence of specific analysis theorems to the axiom of countable choice and examines the axiomatic basis needed for the uniform boundedness principle.

Findings

Arzelà–Ascoli and Fréchet–Kolmogorov theorems are equivalent to countable choice for subsets of reals.

Progress towards understanding the axioms needed for the uniform boundedness principle.

Insights into the foundational requirements of key analysis theorems.

Abstract

In what follows, essentially two things will be accomplished: Firstly, it will be proven that a version of the Arzel\`a--Ascoli theorem and the Fr\'echet--Kolmogorov theorem are equivalent to the axiom of countable choice for subsets of real numbers. Secondly, some progress is made towards determining the amount of axioms that have to be added to the Zermelo--Fraenkel system so that the uniform boundedness principle holds.