Well-ordered set
<mathematics> A set with a total ordering and no infinite descending chains.
A total ordering "<=" satisfies
x <= x
x <= y <= z
=>
x <= z
x <= y <= x
=>
x = y
for all x, y: x <= y or y <= x
In addition, if a set W is well-ordered then all non-empty subsets A of W have a least element, i.e. there exists x in A such that for all y in A, x <= y.
Ordinals are isomorphism classes of well-ordered sets, just as integers are isomorphism classes of finite sets.
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