Axiom of dependent choice

The axiom of dependent choice states that if a binary relation on a nonempty set always gives each element a successor, then an infinite sequence can be formed with each term related to the next.

Serial relation: A binary relation on a set is serial if every element is related to at least one element of that set. Seriality supplies the successor at every stage of the sequence.
Countable choice: Countable choice is the principle that every countable family of nonempty sets has a choice function. Dependent choice implies countable choice by applying its successor rule to finite partial selections.
Axiom of choice: The axiom of choice asserts that every family of nonempty sets has a choice function. The full axiom of choice implies dependent choice, but dependent choice does not imply the full axiom.
Ernst Zermelo: Ernst Zermelo was a German mathematician who formulated the axiom of choice and proved the well-ordering theorem. His work on the axiom of choice established the setting in which weaker choice principles were later studied.
Binary relation: A relation between two sets is a set of ordered pairs; a binary relation on one set relates its elements to one another. The axiom uses a binary relation to specify which consecutive choices are allowed.
Baire category theorem: The Baire category theorem states that a complete metric space is not a countable union of nowhere-dense sets. A standard proof uses dependent choice to construct nested closed sets with shrinking diameters.
Axiom of countable dependent choice: Countable dependent choice is a formulation of dependent choice for successive choices from countable sets. This restricted formulation highlights how the principle handles countable stages and countable choices.
Paul Bernays: Paul Bernays was a Swiss mathematician and logician whose work shaped axiomatic set theory and proof theory. Bernays helped develop the formal study of set-theoretic principles and their relative strength.
Infinite sequence: An infinite sequence is a function whose domain is the natural numbers. The axiom guarantees a sequence indexed by natural numbers, not merely finitely many choices.
Banach–Steinhaus theorem: The Banach–Steinhaus theorem says a pointwise bounded family of continuous linear operators on a Banach space is uniformly bounded. In common foundations, dependent choice supplies the countable recursive constructions used in its proof.

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