Aleph number

:This article is about the aleph numbers. For Aleph One as it pertains to the Marathon computer game, see that article.

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In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph (leph).

Related Topics:
Mathematics - Set theory - Cardinality - Infinite set - Hebrew - Aleph

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The cardinality of the natural numbers is aleph-null (leph_0) (also aleph-naught, aleph-nought); the next larger cardinality is aleph-one leph_1, then leph_2 and so on. Continuing in this manner, it is possible to define a cardinal number leph_lpha for every ordinal number α, as will be described below.

Related Topics:
Natural number - Cardinal number - Ordinal number

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The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

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It should be noted that the aleph numbers are not the same as the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets. Infinity, however, could roughly be defined as the extreme limit of the real number line. While some alephs are larger than others, ∞ is just ∞.

Related Topics:
Infinity - Limit - Real number line

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