Ordinal Multiplication is Closed
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Theorem
Let $x$ and $y$ be ordinals.
Let $\On$ denote the class of all ordinals.
- $x \cdot y \in \On$
Proof
The proof proceeds by transfinite induction on $y$.
Basis for the Induction
| \(\ds x \cdot \O\) | \(=\) | \(\ds \O\) | Definition of Ordinal Multiplication | |||||||||||
| \(\ds \O\) | \(\in\) | \(\ds \On\) | Empty Set is Ordinal | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x \cdot \O\) | \(\in\) | \(\ds \On\) | Substitutivity of Equality |
This proves the basis for the induction.
Induction Step
| \(\ds x \cdot y\) | \(\in\) | \(\ds \On\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {x \cdot y} + y\) | \(\in\) | \(\ds \On\) | Ordinal Addition is Closed | ||||||||||
| \(\ds x \cdot y^+\) | \(=\) | \(\ds \paren {x \cdot y} + y\) | Definition of Ordinal Multiplication | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x \cdot y^+\) | \(\in\) | \(\ds \On\) | Substitutivity of Equality |
This proves the induction step.
Limit Case
| \(\ds \forall z \in y: \, \) | \(\ds x \cdot z\) | \(\in\) | \(\ds \On\) | by hypothesis | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \bigcup_{z \mathop \in y} \paren {x \cdot z}\) | \(\in\) | \(\ds \On\) | Union of Set of Ordinals is Ordinal: Corollary | ||||||||||
| \(\ds x \cdot y\) | \(=\) | \(\ds \bigcup_{z \mathop \in y} \paren {x \cdot z}\) | Definition of Ordinal Multiplication | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x \cdot y\) | \(\in\) | \(\ds \On\) | Substitutivity of Equality |
This proves the limit case.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.16$