For example,

- (said
**aleph-null**or**aleph-naught**) is the cardinality of the set of rational numbers or integers. A set of this cardinality is said to be countably infinite. - (said
**aleph-one**) is the cardinality of the set of all countable ordinal numbers. - is the cardinality of the set of all ordinal numbers of cardinality no greater than aleph-one.

(A more precise definition of the alephs, through all the ordinals, involves the cardinal successor operation).

Table of contents |

2 Aleph-one 3 The continuum hypothesis 4 See also: |

Aleph-null is a transfinite number as defined by Cantor when he proved that infinite sets can have different cardinalities or sizes. Aleph-null is by definition the cardinality of the set of all natural numbers, and is the smallest of all infinite cardinalities. Any set of cardinality Aleph-null can be put into a direct one-to-one correspondence (see bijection) with the integers, and thus is a countably infinite set. Such sets include the set of all prime numbers, the set of all squares of integers, the set of all positive integers, and the set of all integer multiples of a given non-zero real number n.

Aleph-one is the cardinality of the set of all countably infinite ordinal numbers. It can be demonstrated within the Zermelo-Fraenkel axioms (*without* the axiom of choice) that no cardinal number is between aleph-null and aleph-one. If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus aleph-one is the second-smallest infinite cardinal number. Aleph-one is pretty uninteresting without AC; using AC we can show one of the most useful properties of aleph-one: any countable subset of aleph-one has an upper bound in aleph-one (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 (also very useful) fact that any finite subset of aleph-null has an upper bound (finite unions of finite sets are finite).

In Zermelo-Fraenkel set theory *with* the axiom of choice, the celebrated continuum hypothesis is equivalent to the identity