Definition:Cantor Normal Form

Definition
Let $x$ be an ordinal.

Let $k$ be a natural number.

Let $\omega$ denote the minimal infinite successor set.

Let $\langle a_i \rangle$ be a strictly decreasing finite sequence of ordinals.

Let $\langle n_i \rangle$ be a finite sequence of members of $\omega$.

Cantor normal form of $x$ is of the form:


 * $\omega^{a_1} n_1 + \dots + \omega^{a_k} n_k$

In summation notation:


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

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.