Finite Ordinal Times Ordinal

Theorem

Let $m$ and $n$ be finite ordinals.

Let $m \ne 0$, where $0$ is the zero ordinal.

Let $x$ be a limit ordinal.

Then:

$m \times \paren {x + n} = x + \paren {m \times n}$

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Proof

Lemma

Let $m$ be a finite ordinal.

Let $m \ne 0$, where $0$ is the zero ordinal.

Then:

$m \times \omega = \omega$

where $\omega$ denotes the minimally inductive set.

$\Box$

By Ordinal Multiplication is Left Distributive:

$m \times \paren {x + n} = \paren {m \times x} + \paren {m \times n}$

It remains to prove that $x = m \times x$.

Since $x$ is a limit ordinal, it follows that:

\(\ds \exists y \in \On: \, \)

\(\ds x\)

\(=\)

\(\ds \omega \times y\)

Factorization of Limit Ordinals

\(\ds m \times x\)

\(=\)

\(\ds m \times \paren {\omega \times y}\)

Substitutivity of Class Equality

\(\ds m \times x\)

\(=\)

\(\ds \paren {m \times \omega} \times y\)

Ordinal Multiplication is Associative

\(\ds \)

\(=\)

\(\ds \omega \times y\)

lemma

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.29$