In mathematics, the axiom of dependent choice, denoted by , is a weak form of the axiom of choice () that is still sufficient to develop much of real analysis. It was introduced by Paul Bernays in a 1942 article in reverse mathematics that explores which set-theoretic axioms are needed to develop analysis.
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⭐ Core Definition: Axiom of dependent choice
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👉 Axiom of dependent choice in the context of Well-founded
In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset (or subclass) S ⊆ X has a minimal element with respect to R; that is, there exists an m ∈ S such that, for every s ∈ S, one does not have s R m. More formally, a relation is well-founded if:Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.
![{\displaystyle (\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39674b23f300f531f307e8dbab52b953786a5e4b)
Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, meaning there is no infinite sequence x0, x1, x2, ... of elements of X such that xn+1 R xn for every natural number n.