The cardinality of the natural numbers is aleph-null, the next larger cardinality is aleph-one, then comes aleph-two and so on. Continuing in this manner, it is possible to define a cardinal number aleph-κ for every ordinal number κ as will be described below.

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

Table of contents

1 Aleph-null 2 Aleph-one 3 The continuum hypothesis 4 Aleph-two 5 Aleph-ω 6 Aleph-κ for general κ 7 See also:

Aleph-null

Aleph-null (ℵ₀) is by definition the cardinality of the set of all natural numbers, and is the smallest of all infinite cardinalities. A set has cardinality Aleph-null if and only if it is countably infinite, which is the case if and only if it can be put into a direct one-to-one correspondence (see bijection) with the integers. Such sets include the set of all prime numbers and the set of all rational numbers.

Aleph-one

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 in aleph-null (finite unions of finite sets are finite).

The continuum hypothesis

The cardinality of the set of real numbers is 2^aleph-0. It is not clear where this number fits in the aleph number hierarchy. In Zermelo-Fraenkel set theory with the axiom of choice, the celebrated continuum hypothesis is equivalent to the identity

Aleph-two

Aleph-two is the cardinality of the set of all ordinal numbers of cardinality no greater than aleph-one. In ZFC one can prove that aleph-two is the second smallest cardinal number, after aleph-zero and aleph-one.

Aleph-ω

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number aleph-ω is the smallest upper bound of

Aleph-κ for general κ

To define aleph-κ for arbitrary ordinal number κ, we need the successor cardinal operation, which assigns to any cardinal number ρ the next bigger cardinal ρ⁺.

We can then define the aleph numbers as follows