Ordinal Addition
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, ![<span style=]()
{a} union {0,1,2,3,...}" src="https://mathworld.wolfram.com/images/equations/OrdinalAddition/Inline14.gif" style="height:15px; width:119px" />, with 
the least element, is order isomorphic to ![<span style=]()
{0,1,2,3,...}" src="https://mathworld.wolfram.com/images/equations/OrdinalAddition/Inline16.gif" style="height:15px; width:84px" />, but not to ![<span style=]()
{0,1,2,3,...} union {a}" src="https://mathworld.wolfram.com/images/equations/OrdinalAddition/Inline17.gif" style="height:15px; width:119px" />, 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 ![<span style=]()
{alpha+gamma:gamma<beta}" src="https://mathworld.wolfram.com/images/equations/OrdinalAddition/Inline22.gif" style="height:15px; width:80px" /> (Rubin 1967, p. 188; Suppes 1972, p. 205).
REFERENCES:
Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997.
Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.
Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.
Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.
Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.
