In the branch of mathematics known as set theory, 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 
(aleph-null, also aleph-naught, aleph-zero or aleph-nought); 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 a, 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 (8) 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, or an extremal point of the extended real number line. While some alephs are larger than others, 8 is just 8.
Aleph-one
is the cardinality of the set of all countable ordinal numbers, called ?1 or O. Notice ?1 is an uncountable set. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set O (the standard example of a set of size ): any countable subset of O has an upper bound (with respect to the standard well-ordering of ordinals) in O (the proof is easy: a countable union of countable sets is countable; this is one of the most common applications of AC). This fact is analogous to the situation in : any finite set of natural numbers (subset of ?) has a maximum which is also a natural number (has an upper bound in ?) — finite unions of finite sets are finite.
O is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the s-algebra generated by an arbitrary collection of subsets. This is harder than most explicit descriptions of "generation" in algebra (for example vector spaces, groups, etc.) because in those cases we only have to close with respect to finite operations — sums, products, and the like. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of O.