Aleph Number

Aleph Number

In set theory, a discipline within mathematics, the aleph numbers are a sequence 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 .

The cardinality of the natural numbers is (read aleph-naught, aleph-null, or aleph-zero), the next larger cardinality is aleph-one, then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number α, as described below.

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

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.