Aleph Null¶

Definition¶

Aleph null, denoted $\aleph_0$, is the cardinality of the natural numbers:

Interpretation¶

$\aleph_0$ is the name for the size of the set of natural numbers. A set has this size exactly when its elements can be listed as a sequence $a_0,a_1,a_2,\ldots$ in which every element appears exactly once, that is, when it is countably infinite.

The figure pairs each natural number $n$ with $2n$. Every natural number gets exactly one partner and every even number is hit exactly once, so $n\mapsto2n$ is a bijection from $\mathbb{N}$ onto the even numbers $2\mathbb{N}$. Hence $2\mathbb{N}$ also has cardinality $\aleph_0$, although it is a proper subset of $\mathbb{N}$ that misses every odd number. For infinite sets, being a part does not mean being smaller.

The same listing idea explains the cardinal arithmetic $\aleph_0+1=\aleph_0+\aleph_0=\aleph_0\cdot\aleph_0=\aleph_0$${\aleph }_0$. Adding one element $x$ to $\mathbb{N}$ still gives a list $x,0,1,2,\ldots$. Two disjoint countably infinite sets can be listed together by taking elements from each in turn, and $\mathbb{N}\times\mathbb{N}$ can be listed along the finite diagonals $i+j=0,1,2,\ldots$$0,1,2,\dots$. Ideas of this kind show that $\mathbb{Z}$ and $\mathbb{Q}$ have cardinality $\aleph_0$ as well.

Not every infinite set has this size: by Cantor's diagonal argument the real numbers cannot be listed, and the cardinality of the continuum is strictly larger. So $\aleph_0$ is one particular infinite size, not the same as the symbol $\infty$ from calculus, which describes unbounded growth rather than the size of a set.

Key Results¶

• A set has cardinality $\aleph_0$ exactly when it is countably infinite.
• A bijection is the main tool for proving that two sets have the same cardinality.

Examples¶

• $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ all have cardinality $\aleph_0$.

Prerequisites¶

• Natural Number
• Cardinality

Related Concepts¶

• Countable and Uncountable Sets
• Power Set
• Cantor's Theorem
• Cantor's Diagonal Argument
• Cardinality of the Continuum

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: aleph-nought, aleph-zero, aleph 0, aleph-naught Keywords: countable infinity Symbols: infinity, integers, natural numbers, rational numbers

Last updated: 20 October 2026 03:02