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.

By disjunctive syllogism:

$y + x \in K_{II}$

$\blacksquare$


Sources