Ordinal exponentiation ${\alpha }^{\beta }$, with the conventions ${\alpha }^0=1$ and $0^0=1$

Let $\alpha$ and $\beta$ be ordinals (Ordinal (von Neumann)). The power ${\alpha }^{\beta }$ is defined by recursion on $\beta$, in the three cases of Successor and limit ordinals:

${\alpha }^0=1,{\alpha }^{{\beta }^{+}}={\alpha }^{\beta }\cdot \alpha ,{\alpha }^{\lambda }=\bigcup \{{\alpha }^{\beta }:0<\beta <\lambda \}(\lambda \text{ a limit ordinal}),$

with $\cdot$ the ordinal multiplication of Ordinal multiplication $\alpha \cdot \beta$. That exactly one operation satisfies these three clauses, and that all its values are ordinals, is Ordinal exponentiation exists and is unique, with the limit clause taken over $0<\beta <\lambda$ so that $0^{\lambda }=0$, proved immediately above.

The first clause applies to every $\alpha$, so in particular $0^0=1$.

• The limit clause ranges over $0<\beta <\lambda$, not over $\beta <\lambda$, and the restriction is load bearing. The unrestricted union would include the value ${\alpha }^0=1$, and at $\alpha =0$ that single stray term flips the answer: it would make $0^{\omega }=1$ instead of $0$. With the restriction, one formula is correct for every $\alpha$ at once and no case split on $\alpha$ is needed. Ordinal exponentiation exists and is unique, with the limit clause taken over $0<\beta <\lambda$ so that $0^{\lambda }=0$ carries the details of that restriction; ${\alpha }^{\beta +\gamma }={\alpha }^{\beta }\cdot {\alpha }^{\gamma }$ and $({\alpha }^{\beta }{)}^{\gamma }={\alpha }^{\beta \cdot \gamma }$; and for $\alpha >1$ exponentiation is strictly increasing with $\beta \le {\alpha }^{\beta }$ proves the exponent law that the unrestricted clause would falsify. For $\alpha \ge 1$ the two clauses agree, because then ${\alpha }^0=1\le \alpha ={\alpha }^1$ and $1<\lambda$, so dropping the term at $\beta =0$ does not lower the supremum.

• ${\alpha }^1=\alpha$. Indeed ${\alpha }^1={\alpha }^{0^{+}}={\alpha }^0\cdot \alpha =1\cdot \alpha$, and $1\cdot \alpha =\alpha$ is proved in Monotonicity of ordinal $+$ and $\cdot$: strictly increasing and continuous in the right argument, weakly increasing in the left, with left cancellation, and the identities $0+\beta =\beta$ and $1\cdot \beta =\beta$.

• This is not cardinal exponentiation. The ordinal $2^{\omega }$ is $\omega$, computed in FALSE: the ordinal $2^{\omega }$ is uncountable; the cardinal power of $2$ by the size of $\mathbb{N}$ is the size of $\mathcal{P}(\mathbb{N})$, which is uncountable. The two operations share one notation and are not the same function. Ordinal ${\alpha }^{\beta }$ and cardinal ${\kappa }^{\lambda }$ are different operations that share one notation is the standing warning, and cardinal exponentiation is not defined at this point in the reading order; it is introduced later, on Cardinal Arithmetic, Cofinality and the Alephs.

• Notation and precedence. ${\alpha }^{\beta }\cdot \gamma$ means $({\alpha }^{\beta })\cdot \gamma$, and ${\alpha }^{\beta }+\gamma$ means $({\alpha }^{\beta })+\gamma$; powers bind tightest, then products, then sums. The Cantor normal form of Cantor normal form: every nonzero ordinal is ${\omega }^{{\beta }_0}\cdot c_0+\cdots +{\omega }^{{\beta }_{k-1}}\cdot c_{k-1}$ with ${\beta }_0>\cdots >{\beta }_{k-1}$ and each $c_i$ a nonzero natural number, in exactly one way is written with that convention throughout.

• The base is a parameter, the exponent is what the recursion runs on. As with Ordinal addition $\alpha +\beta$ and Ordinal multiplication $\alpha \cdot \beta$, the recursion is on the right argument, and the operation is correspondingly asymmetric: $\alpha \mapsto {\alpha }^{\beta }$ and $\beta \mapsto {\alpha }^{\beta }$ behave quite differently, and only the second is continuous at limits.