Limit Ordinals Preserved Under Ordinal Addition
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Theorem
Let $x$ and $y$ be ordinals such that $x$ is a limit ordinal.
Then $y + x$ is a limit ordinal.
That is, letting $K_{II}$ denote the class of all limit ordinals:
- $\forall x \in K_{II}: y + x \in K_{II}$
Proof
\(\ds x\) | \(\in\) | \(\ds K_{II}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\ne\) | \(\ds \O\) | Definition of Limit Ordinal | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y + x\) | \(\ne\) | \(\ds \O\) | Ordinal is Less than Sum | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y + x\) | \(\in\) | \(\ds K_{II}\) | Definition of Limit Ordinal | ||||||||||
\(\, \ds \lor \, \) | \(\ds y + x\) | \(=\) | \(\ds z^+\) |
The result is now obtained by Proof by Contradiction:
Assume that $y + x = z^+$.
\(\ds y + x\) | \(=\) | \(\ds z^+\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \bigcup_{w \mathop \in x} \paren {y + w}\) | \(=\) | \(\ds z^+\) | Definition of Ordinal Addition | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(\in\) | \(\ds \bigcup_{w \mathop \in x} \paren {y + w}\) | Ordinal is Less than Successor | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists w \in x: \, \) | \(\ds z\) | \(\in\) | \(\ds y + w\) | Definition of Set Union |
But $w \in x \implies w^+ \in x$ by Successor of Ordinal Smaller than Limit Ordinal is also Smaller.
\(\ds z\) | \(\in\) | \(\ds y + w\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^+\) | \(\in\) | \(\ds \paren {y + w}^+\) | Successor is Less than Successor | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^+\) | \(\in\) | \(\ds y + w\) | Definition of Ordinal Addition | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^+\) | \(\in\) | \(\ds z^+\) | by hypothesis |
But $z^+ \in z^+$ is clearly a membership loop, and therefore, our assumption must be wrong.
- $y + x \in K_{II}$
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.11$