Let
and
be disjoint totally ordered sets with order
types
and .
Then the ordinal sum is defined at set where, if and are both from the same subset,
the order is the same as in the subset, but if is from and is from , then has order type (Ciesielski 1997, p. 48;
Dauben 1990, p. 104; Moore 1982, p. 40).

One should note that in the infinite case, order type
addition is not commutative, although it is associative. For example,

(1)

In addition, ,
with
the least element, is order isomorphic to , but not to , with the greatest element, since it has a greatest element and
the other does not.

An inductive definition for ordinal addition states that for any ordinal number ,

(2)

and

(3)

If
is a limit ordinal, then is the least ordinal greater than any ordinal in
the set
(Rubin 1967, p. 188; Suppes 1972, p. 205).