Ordinal $\alpha^{\beta}$ and cardinal $\kappa^{\lambda}$ are different operations that share one notation

The notation $\alpha^{\beta}$ is used in set theory for two different operations, and on this page it always means the first of them.

Ordinal exponentiation, the one defined here (Ordinal exponentiation $\alpha^{\beta}$, with the conventions $\alpha^{0} = 1$ and $0^{0} = 1$), is built by transfinite recursion on the exponent, with a supremum at limits. Its value depends on the ordinals $\alpha$ and $\beta$ as order types, and the operation is designed so that $\beta \mapsto \alpha^{\beta}$ is strictly increasing and continuous for $\alpha > 1$.

Cardinal exponentiation is a different operation, defined on cardinals (Cardinal (initial ordinal) and cardinality) by counting functions: $\kappa^{\lambda}$ is the number of functions from a set of size $\lambda$ to a set of size $\kappa$. It is not defined at this point in the reading order; it is introduced later, on Cardinal Arithmetic, Cofinality and the Alephs. It is named in this remark only to warn the reader off the identification.

The two disagree at the smallest interesting input. As ordinals,

$2^{\omega} = \omega,$

computed in FALSE: the ordinal $2^{\omega}$ is uncountable from the limit clause: every $2^{n}$ with $n \in \omega$ is again a natural number, so the supremum of the tower is $\omega$ itself, and the result is countably infinite (Finite, countably infinite, countable, uncountable). The cardinal reading of the same symbols asks instead for the number of functions $\mathbb{N} \to \{0,1\}$, that is for the size of $\mathcal{P}(\mathbb{N})$, and $\mathcal{P}(\mathbb{N})$ is uncountable: there is no surjection $\mathbb{N} \to \mathcal{P}(\mathbb{N})$ at all, by Cantor's theorem: $A \prec \mathcal{P}(A)$. So under one reading the answer is the smallest infinite ordinal, and under the other it is a set strictly larger than $\mathbb{N}$.

Why this remark is here rather than in a footnote. A reader who knows that "$2^{\aleph_0}$ is uncountable" and then meets $2^{\omega}$ on this page has every reason to expect an uncountable ordinal, and would conclude that something above has gone wrong. Nothing has: the two expressions are values of two different functions. This page writes $\omega$ for the least infinite ordinal throughout and $\omega_1$ for the first uncountable one, and never writes $\aleph_0$ or $\aleph_1$, precisely so that an ordinal expression here is never silently read as a cardinal one. Where the aleph subscript notation appears elsewhere in this library it is inside a statement about cardinal arithmetic, never inside an ordinal computation; no page of the ordinal development uses it.

What else is nearby, and what it is not. $\mathbb{R}$ is uncountable too ($\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)), by an argument that has nothing to do with power sets; and whether any set sits strictly between $\mathbb{N}$ and $\mathcal{P}(\mathbb{N})$ in size is the continuum hypothesis, independent of ZFC (The continuum hypothesis, and what this page does not prove). None of that is a statement about ordinal arithmetic, and none of it bears on the value $2^{\omega} = \omega$ proved on this page.

A rule of thumb that is safe here. If the exponent is being used to index a transfinite recursion, the exponentiation is ordinal. If it is being used to count functions, it is cardinal. On this page it is always the first, because the second is not defined at this point in the reading order.