Aleph Null ($\aleph_0$)

Aleph null (or aleph naught), denoted $\aleph_0$, is the cardinality of the natural numbers $\mathbb{N}$.

Definition

$$ \aleph_0 = |\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}| $$

A set has cardinality $\aleph_0$ if and only if it is countably infinite—that is, its elements can be put into a one-to-one correspondence with $\mathbb{N}$.

Properties

• $\aleph_0$ is the smallest infinite cardinal
• $\aleph_0 + \aleph_0 = \aleph_0$
• $\aleph_0 \cdot \aleph_0 = \aleph_0$
• $2^{\aleph_0} = |\mathbb{R}| = \mathfrak{c}$ (the cardinality of the continuum)

Examples of Sets with Cardinality $\aleph_0$

• Natural numbers $\mathbb{N}$
• Integers $\mathbb{Z}$
• Rational numbers $\mathbb{Q}$
• Algebraic numbers
• Any infinite subset of a countable set

Related

• cardinality
• countable
• bijection