Ordinal Addition

Let (A,<=) and (B,<=) be disjoint totally ordered sets with order types alpha and beta. Then the ordinal sum is defined at set (C=A union B,<=) where, if c_1 and c_2 are both from the same subset, the order is the same as in the subset, but if c_1 is from A and c_2 is from B, then c_1<c_2 has order type alpha+beta (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,


In addition, {a} union {0,1,2,3,...}, with a the least element, is order isomorphic to {0,1,2,3,...}, but not to {0,1,2,3,...} union {a}, with a 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 alpha,



 alpha+(successor to beta)=the successor to (alpha+beta).

If beta is a limit ordinal, then alpha+beta is the least ordinal greater than any ordinal in the set {alpha+gamma:gamma<beta} (Rubin 1967, p. 188; Suppes 1972, p. 205).

See also

Ordinal Exponentiation, Ordinal Multiplication, Ordinal Number

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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.

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Ordinal Addition

Cite this as:

Weisstein, Eric W. "Ordinal Addition." From MathWorld--A Wolfram Web Resource.

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