Well-founded relation
In mathematics, a binary relation,
R, is
well-founded on a class
X if and only if every non-empty subset
S⊆X has a minimal element; that is, some element
m of any
S is not related by
sRm for the rest of the
s ∈ S. Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descending chains: that is, there is no infinite sequence
x0,
x1,
x2,... of elements of
X such that
xn+1
R xn for every natural number
n. In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order. In set theory, a set
x is called a
well-founded set if the set membership relation is well-founded on the transitive closure of
x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded.