Aleph-null

Description

Aleph-null, written $\aleph_0$, is the cardinality of the set $\mathbb{N}$ of natural numbers. It is the cardinality of each countably infinite set. It is the least infinite cardinal number: each infinite set has cardinality at least $\aleph_0$, with the axiom of choice.

Remarks

Other names are aleph-zero and aleph-nought. $\aleph_0 + \aleph_0 = \aleph_0 \cdot \aleph_0 = \aleph_0$, but $2^{\aleph_0} > \aleph_0$ by Cantor's theorem. As an ordinal, $\aleph_0$ is $\omega$, the set of the finite ordinals.

Explanations

L1 human

$\aleph_0$ is the size of the infinity of counting. The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ all have this size.