A cardinal number, or cardinal, is a kind of number that measures the cardinality of a set, meaning how many elements the set contains; two sets share a cardinal number exactly when there exists a one-to-one correspondence, or bijection, between their elements. For a finite set, the cardinal number can be identified with an ordinary natural number found simply by counting, but the behavior of cardinality among infinite sets is more complex: for instance the natural numbers and the rational numbers can be placed in bijection with each other despite the natural numbers forming a proper subset of the rationals, while Georg Cantor showed that the real numbers have a strictly larger cardinality than the natural numbers. The cardinality of the natural numbers is written aleph-null, the smallest of the aleph numbers, and whether every infinite cardinal is an aleph number depends on the axiom of choice.
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1. Wikipedia: Cardinal number
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A cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set.
In Branch: Set Theory, Lead paragraph
The properties of other aleph numbers and of infinite cardinal numbers in general depend on statements independent of Zermelo-Fraenkel set theory, such as the axiom of choice and the continuum hypothesis.
Attributed To: Georg Cantor, Lead
The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874-1884
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