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cardinal number

A number, often called simply a cardinal, that is used to count the objects or ideas in a set or collection: zero, one, two, ... , eighty-three, and so on. The cardinality of a set is just the number of elements the set contains. For finite sets this is always a natural number. To compare the sizes of two sets, X and Y, all that's necessary is to pair off the elements of X with those of Y and see if there are any left over. This concept is obvious in the case of finite sets but leads to some strange conclusions when dealing with infinite sets (see infinity). For example, it is possible to pair off all the natural numbers with all the even numbers, with none left over; thus the set of natural numbers and the set of even numbers have the same cardinality. In fact, an infinite set can be defined as any set that has a proper subset of the same cardinality. Every countable set that is infinite has a cardinality of aleph-null; the set of real numbers has cardinality aleph-one. See also ordinal number.

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