# Definition:Cantor Normal Form

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## Definition

Let $x$ be an ordinal.

The **Cantor normal form** of $x$ is an ordinal summation:

- $x = \omega^{a_1} n_1 + \dots + \omega^{a_k} n_k$

where:

- $k \in \N$ is a natural number
- $\omega$ is the minimal infinite successor set
- $\langle a_i \rangle$ is a strictly decreasing finite sequence of ordinals.
- $\langle n_i \rangle$ is a finite sequence of finite ordinals

In summation notation:

- $x = \displaystyle \sum_{i \mathop = 1}^k \omega^{a_i} n_i$

## Properties

Every ordinal number can be written in **Cantor normal form**.

Moreover, the Cantor normal form is unique.

The ordinal cannot be written any other way that could still be considered **Cantor normal form**.

This unique representation is a consequence of the Division Theorem for Ordinals.

Cantor normal form is useful when performing operations like multiplication and exponentiation.

See Ordinal Multiplication via Cantor Normal Form/Limit Base and Ordinal Exponentiation via Cantor Normal Form/Limit Exponents.

## Also see

- Unique Representation of Ordinal as Sum shows that
**Cantor normal form**exists for every ordinal and is unique.

## Source of Name

This entry was named for Georg Cantor.