# 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 minimally inductive set
- $\sequence {a_i}$ is a strictly decreasing finite sequence of ordinals.
- $\sequence {n_i}$ is a finite sequence of finite ordinals

In summation notation:

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

This article, or a section of it, needs explaining.In particular: It still needs to be explained why, when used in pages that link to this, that the summation does not include the object $\omega$ in it, just some ordinal $x$ instead. It is unclear exactly what this definition means, because $\omega$, as currently defined on this website, is the Definition:Minimally Inductive Set. Thus this definition appears to be saying: "Every ordinal (which of course has to include finite ones) can be expressed as finite sums of infinite ordinals." How can a finite number (an ordinal is a number, right?) be expressed as the sum of infinite numbers?You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

## Properties

This page or section has statements made on it that ought to be extracted and proved in a Theorem page.In particular: Remove this section and replace with links to results stating and proving these statements, and reference them (if considered necessary) in "Also see".You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.To discuss this page in more detail, feel free to use the talk page. |

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.