On the logical structure of some maximality and well-foundedness principles equivalent to choice principles

TL;DR

This paper explores the logical structure of maximality and well-foundedness principles, showing their equivalence to choice principles and generalizing classical results like Zorn's lemma within a formal mathematical framework.

Contribution

It establishes the equivalence between the Teichmüller-Tukey lemma, update induction, and other choice-related principles, extending classical theorems to broader contexts.

Findings

Teichmüller-Tukey lemma is equivalent to the axiom of choice.

Generalized forms of maximality and well-foundedness principles are equivalent to choice.

Includes a variant of Zorn's lemma and compares with existing choice and bar induction principles.

Abstract

We study the logical structure of Teichm{\"u}ller-Tukey lemma, a maximality principle equivalent to the axiom of choice and show that it corresponds to the generalisation to arbitrary cardinals of update induction, a well-foundedness principle from constructive mathematics classically equivalent to the axiom of dependent choice.From there, we state general forms of maximality and well-foundedness principles equivalent to the axiom of choice, including a variant of Zorn's lemma. A comparison with the general class of choice and bar induction principles given by Brede and the first author is initiated.