well-ordered set
A set with a {total ordering} and no infinite
descending {chain}s. A total ordering "<◦" satisfies x <◦ x;
x <◦ y <◦ z ◦> x <◦ z; x <◦ y <◦ x ◦> x◦y; and 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.
{Ordinal}s are {isomorphism class}es of {well-ordered set}s,
just as {integer}s are {isomorphism class}es of finite sets.
(1995-04-19)