On the logical structure of some maximality and well-foundedness principles equivalent to choice principles
Hugo Herbelin
TL;DR
This work analyzes Teichmüller-Tukey lemma as a maximality principle equivalent to the axiom of choice and reveals a constructive duality with update induction, extending the classical equivalences to arbitrary cardinals. It develops a unified framework connecting finite-character predicates, open predicates, and their duals, and shows TTL and Zorn's lemma are mutually intertransformable within this setting. The paper further introduces ∃MPCF and links it to GDC, demonstrating that these maximality principles precisely capture choice-like strength and clarifying how variants restricted to countable domains or two-valued codomains align with dependent choice. Overall, it bridges maximality, well-foundedness, and the broader family of choice/bar-induction principles, offering new avenues for analysis and comparison in constructive logic. The results illuminate how maximality principles can encode totality conditions and how these insights inform the classification of related induction principles.
Abstract
We study the logical structure of Teichm{ü}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.