Ordinal Multiplication by Zero

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Theorem

Let $x$ be an ordinal.

\(\displaystyle \left({x \cdot \varnothing}\right)\) \(=\) \(\displaystyle \varnothing\)
\(\displaystyle \left({\varnothing \cdot x}\right)\) \(=\) \(\displaystyle \varnothing\)


Proof

\(\displaystyle \left({x \cdot \varnothing}\right)\) \(=\) \(\displaystyle \varnothing\) Definition of Ordinal Multiplication

For $\left({\varnothing \cdot x}\right) = \varnothing$, the proof shall proceed by Transfinite Induction on $x$.


Basis for the Induction

\(\displaystyle \left({\varnothing \cdot \varnothing}\right)\) \(=\) \(\displaystyle \varnothing\) Definition of Ordinal Multiplication

This proves the basis for the induction.


Induction Step

\(\displaystyle \left({\varnothing \cdot x}\right)\) \(=\) \(\displaystyle \varnothing\) Inductive Hypothesis
\(\displaystyle \implies \ \ \) \(\displaystyle \left({\left({\varnothing \cdot x}\right) + \varnothing}\right)\) \(=\) \(\displaystyle \varnothing\) Definition of Ordinal Addition
\(\displaystyle \left({\varnothing \cdot x^+}\right)\) \(=\) \(\displaystyle \left({\left({\varnothing \cdot x}\right) + \varnothing}\right)\) Definition of Ordinal Multiplication
\(\displaystyle \implies \ \ \) \(\displaystyle \left({\varnothing \cdot x^+}\right)\) \(=\) \(\displaystyle \varnothing\) Equality is Transitive

This proves the induction step.


Limit Case

\(\, \displaystyle \forall y \in x: \, \) \(\displaystyle \left({\varnothing \cdot y}\right)\) \(=\) \(\displaystyle \varnothing\) Hypothesis
\(\displaystyle \implies \ \ \) \(\displaystyle \bigcup_{y \mathop \in x} \left({\varnothing \cdot y}\right)\) \(=\) \(\displaystyle \varnothing\) Indexed Union Equality
\(\displaystyle \left({\varnothing \cdot x}\right)\) \(=\) \(\displaystyle \bigcup_{y \mathop \in x} \left({\varnothing \cdot y}\right)\) Definition of Ordinal Multiplication
\(\displaystyle \implies \ \ \) \(\displaystyle \left({\varnothing \cdot x}\right)\) \(=\) \(\displaystyle \varnothing\) Equality is Transitive

This proves the limit case.

$\blacksquare$


Sources