Limit Ordinals Closed under Ordinal Exponentiation

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Let $x$ and $y$ be ordinals.

Let $y$ be a limit ordinal.

Let $x^y$ denote ordinal exponentiation.


If $x > 1$, then $x^y$ is a limit ordinal.
If $x \ne \O$, then $y^x$ is a limit ordinal.


Suppose $x > 1$.

Suppose also that $x^y$ is the successor of some ordinal $w$.

By definition of ordinal multiplication:

$\ds x^y = \bigcup_{z \mathop \in y} x^z$


\(\ds w\) \(\in\) \(\ds x^y\) Ordinal is Less Than Successor
\(\ds \exists z \in y: \, \) \(\ds w\) \(\in\) \(\ds x^z\) Definition of Ordinal Exponentiation
\(\ds w^+\) \(\subseteq\) \(\ds x^z\) Successor of Element of Ordinal is Subset
\(\ds \) \(\in\) \(\ds x^{z^+}\) Membership is Left Compatible with Ordinal Exponentiation

But $z^+ \in y$ by Successor of Ordinal Smaller than Limit Ordinal is also Smaller.

So $w^+ \in x^y$ and $w^+ \in w^+$.

This creates a membership loop and thus is a contradiction by No Membership Loops.


For the second part, since $x$ is not the empty set it follows that $x = z^+$ for some $z$ or that $x$ is a limit ordinal.

If $x$ is a limit ordinal, then $y^x$ is a limit ordinal by the first part.

If $x$ is the successor of $z$, then:

$y^x = y^w × y$ by the definition of ordinal exponentiation.

Then, $y^x$ is a limit ordinal by Limit Ordinals Preserved Under Ordinal Multiplication.