Wellfoundedness

mathematical concept generalizing ordinal numerals to extend enumeration to infinite sets

In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then called a well-ordered set (or woset). In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering.

type of binary relation

axiom stating that all sets are well-founded

in set theory, a set which contains all objects, including itself

form of mathematical proof

Statement that all sets of positive numbers contains a least element

lemma in infinite graph theory

condition in commutative algebra

well-quasi-ordering of finite trees

theorem

topological space with no infinite strictly ascending sequence of closed subsets

mathematical theorem

variants of axiomatic set theory that allow sets to be elements of themselves

lemma

In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X is a quasi-ordering of X for which every infinite sequence of elements x_0, x_1, x_2, \ldots from X contains a non-decreasing pair x_i \leq x_j with i

In set theory, \in-induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction.