Aleph number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

The smallest cardinality of an infinite set is that of the natural numbers, denoted by

ℵ

0

{\displaystyle \aleph _{0}}

(read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-ordered set is

ℵ

1

,

{\displaystyle \aleph _{1},}

then

ℵ

2

,

{\displaystyle \aleph _{2},}

then

ℵ

3

,

{\displaystyle \aleph _{3},}

and so on. Continuing in this manner, it is possible to define an infinite cardinal number

ℵ

α

{\displaystyle \aleph _{\alpha }}

for every ordinal number

α

,

{\displaystyle \alpha ,}

as described below.

The concept and notation are due 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 (

∞

{\displaystyle \infty }

) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.

Aleph-zero

ℵ

0

{\displaystyle \aleph _{0}}

(aleph-nought, aleph-zero, or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called

ω

{\displaystyle \omega }

or

ω

0

{\displaystyle \omega _{0}}

(where

ω

{\displaystyle \omega }

is the lowercase Greek letter omega), also has cardinality

ℵ

0

{\displaystyle \aleph _{0}}

. A set has cardinality

ℵ

0

{\displaystyle \aleph _{0}}

if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are:

the set of natural numbers, irrespective of including or excluding zero,

the set of all integers,

any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers,

the set of all rational numbers,

the set of all constructible numbers (in the geometric sense),

the set of all algebraic numbers,

the set of all computable numbers,

the set of all computable functions,

the set of all binary strings of finite length, and

the set of all finite subsets of any given countably infinite set.