Cardinal Number
In informal usage, a cardinal number is a number used in counting (a Counting Number), such as 1, 2, 3, ....
Formally, a cardinal number is a type of number defined in such a way that any method of counting Sets usingit gives the same result. (This is not true for the Ordinal Numbers.) In fact, the cardinalnumbers are obtained by collecting all Ordinal Numbers which are obtainable by counting a givenset. A set has 
(Aleph-0) members if it can be put into a One-to-One correspondence with the finiteOrdinal Numbers.
Two sets are said to have the same cardinal number if all the elements in the sets can bepaired off One-to-One. An Inaccessible Cardinal cannot be expressed in terms of a smaller number of smallercardinals.
See also Aleph, Aleph-0, Aleph-1, Cantor-DedekindAxiom, Cantor Diagonal Slash, Continuum, Continuum Hypothesis, Equipollent, InaccessibleCardinals Axiom, Infinity, Ordinal Number, Power Set, Surreal Number, Uncountable Set
References
Cantor, G. Über unendliche, lineare Punktmannigfaltigkeiten, Arbeiten zur Mengenlehre aus dem Jahren 1872-1884. Leipzig, Germany: Teubner, 1884. Conway, J. H. and Guy, R. K. ``Cardinal Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 277-282, 1996. Courant, R. and Robbins, H. ``Cantor's `Cardinal Numbers.''' §2.4.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 83-86, 1996.
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