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In mathematics, aleph, the first letter of the Hebrew alphabet, in its uppercase form ( ) denotes numbers representing the sizes of infinite set s. Such numbers are known as transfinite cardinal number s, or transfinite cardinals.

The smallest transfinite cardinal is , called aleph null or aleph nought. This is the cardinality of (the number of elements in) the set of natural number s. It is also the cardinality of the set of integer s and the set of rational number s. This number is denumerable, in the sense that the elements of any set with cardinality can be entirely defined by writing a list, or at least the first few elements of a list. For example, the set of integers can be listed as {0, 1, -1, 2, -2, 3, -3, ...}. Although it is not possible to completely write out this list, its elements are implied by the continuation symbol (...).

The cardinality of the set of real number s is larger than . The real numbers cannot be denoted by any sort of list. This was proven by the German mathematician Georg Cantor (1845-1918), and produced the counterintuitive result that some infinities are larger than others.

See Mathematical Symbols .

This was last updated in September 2005

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