Cardinal Addition
Let 
and 
be any sets with empty intersection, and let 
denote the cardinal number of a set 
. Then
(Ciesielski 1997, p. 68; Dauben 1990, p. 173; Rubin 1967, p. 274; Suppes 1972, pp. 112-113).
It is an interesting exercise to show that cardinal addition is well-defined. The main steps are to show that for any cardinal numbers 
and 
, there exist disjoint sets 
and 
with cardinal numbers 
and 
, and to show that if 
and 
are disjoint and 
and 
disjoint with 
and 
then 
. The second of these is easy. The first is a little tricky and requires an appeal to the axioms of set theory. Also, one needs to restrict the definition of cardinal to guarantee if 
is a cardinal, then there is a set 
satisfying 
.
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.
Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.
Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.
