Ordinal Multiplication by One
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Theorem
Let $x$ be an ordinal.
Let $1$ denote the ordinal one.
| \(\ds x \cdot 1\) | \(=\) | \(\ds x\) | ||||||||||||
| \(\ds 1 \cdot x\) | \(=\) | \(\ds x\) |
Proof
| \(\ds x \cdot 1\) | \(=\) | \(\ds x \cdot \O^+\) | Definition of One (Ordinal) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {x \cdot \O} + x\) | Definition of Ordinal Multiplication | |||||||||||
| \(\ds \) | \(=\) | \(\ds \O + x\) | Definition of Ordinal Multiplication | |||||||||||
| \(\ds \) | \(=\) | \(\ds x\) | Ordinal Addition by Zero |
$\Box$
The proof of the other equality shall proceed by Transfinite Induction.
Basis for the Induction
| \(\ds 1 \cdot \O\) | \(=\) | \(\ds \O\) | Definition of Ordinal Multiplication |
This proves the basis for the induction.
Induction Step
| \(\ds 1 \cdot x\) | \(=\) | \(\ds x\) | Inductive Hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {1 \cdot x} + 1\) | \(=\) | \(\ds x^+\) | Ordinal Addition by One | ||||||||||
| \(\ds 1 \cdot x^+\) | \(=\) | \(\ds \paren {1 \cdot x} + 1\) | Definition of Ordinal Multiplication | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1 \cdot x^+\) | \(=\) | \(\ds x^+\) | Equality is Transitive |
This proves the induction step.
Limit Case
| \(\ds \forall y \in x: \, \) | \(\ds 1 \cdot y\) | \(=\) | \(\ds y\) | by hypothesis | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \bigcup_{y \mathop \in x} \paren {1 \cdot y}\) | \(=\) | \(\ds \bigcup_{y \mathop \in x} y\) | Indexed Union Equality | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1 \cdot x\) | \(=\) | \(\ds \bigcup_{y \mathop \in x} y\) | Definition of Ordinal Multiplication | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1 \cdot x\) | \(=\) | \(\ds x\) | Limit Ordinal Equals its Union |
This proves the limit case.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.18$