Limit Ordinals Closed under Ordinal Exponentiation

Theorem

Let $x$ and $y$ be ordinals.

Let $y$ be a limit ordinal.

Then:

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 $w$.

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

Then:

$\ds w$

$\in$

$\ds x^y$

$\ds \exists z \in y: \, $

$\ds w$

$\in$

$\ds x^z$

$\ds w^+$

$\subseteq$

$\ds x^z$

$\ds $

$\in$

$\ds x^{z^+}$

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

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

$\Box$

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.

$\blacksquare$