Seeing how the essential question was answered, I want to stress something else in your post which needs to be pointed out:
It is true that cardinals (namely, Aleph numbers) are usually treated as ordinals, however the multiplications and addition of cardinals and ordinals are very different, and most of all - exponentiation is different as well.
For ordinals $\alpha$ and $\beta$ we define the sum to be:
- $\alpha + 0 = \alpha$
- $\alpha + (\beta + 1) = (\alpha + \beta) + 1$ (where $+1$ is the successor ordinal)
- $\alpha + \beta$ for a limit ordinal $\beta$ is the limit of $\alpha+\gamma$ for $\gamma<\beta$
One can notice that it is usually non-commutative as $2+\omega = \sup\{2+n\colon n<\omega\} = \omega \neq \omega+2$.
The ordinal multiplication is defined in a similar way, as well exponentiation. (Namely a simple rule for zero, and successor and a limit for limits) and an interesting result is that $\omega^\omega$ is countable when dealing with ordinal exponentiation.
In contrast, if $\lambda$ and $\mu$ are cardinals then $\lambda + \mu = \mu + \lambda = \lambda \cdot \mu = \mu \cdot \lambda = \max \{\lambda, \mu\}$, and the exponentiation is defined as $\lambda^\mu = |\{f | f\colon\mu\to\lambda\}|$ - that is the cardinality of the collection of functions from $\mu$ into $\lambda$.
For further information and definitions you can see this wikipedia link
So once you were dealing with the cardinality of the basis you were looking for cardinal arithmetics and not ordinal arithmetics. Which are two different things.
