Well-founded in the context of Subset

⭐ Core Definition: 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) $S \subseteq X$ has a minimal element with respect to $R$ ; that is, there exists an $m \in S$ such that, for every $s \in S$ , one does not have $s R m$ . More formally, a relation is well-founded if: $(\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].$ Some authors include an extra condition that $R$ is set-like, i.e., that the elements less than any given element form a set.

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 $x 0, x 1, x 2, ...$ of elements of $X$ such that $x n +1 R x n$ for every natural number $n$ .

↓ Menu

Well-founded in the context of Zermelo–Fraenkel set theory

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.

View the full Wikipedia page for Zermelo–Fraenkel set theory

↑ Return to Menu

Well-founded Study page number 1 of 1

Play TriviaQuestions Online!

Skip to study material about Well-founded in the context of "Subset"

⭐ Core Definition: Well-founded

Well-founded in the context of Zermelo–Fraenkel set theory