Definition:Cantor Normal Form

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

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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.