well-ordered set
TOTAL ORDER SUCH THAT EVERY NONEMPTY SUBSET OF THE DOMAIN HAS A LEAST ELEMENT
Well-ordered set; Well-ordered; Well-ordering; Well ordered; Well ordering; Well-ordering property; Wellorder; Wellordering; Well ordered set; Wellordered; Well ordering theory; Well ordering property; Well-Ordering; Well-Ordered; Well-orderable set; Well order
<
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.
(1995-04-19)