The reason is that the definitions of ordinal arithmetics and cardinal arithmetics are very different.
Whereas the ordinal arithmetics operations are concerned with order types, the cardinal arithmetics are concerned with certain sets.
For example, $\alpha+\beta$ as ordinals is the order type of $\alpha$ concatenated with $\beta$. Whereas cardinality strips naked any possible structure, and considering the cardinality of the set $\{0\}\times\alpha\cup\{1\}\times\beta$, which is equal to the maximum of $|\alpha|$ and $|\beta|$ (granted one is infinite).
Exponentiation, which is the strangest one, is defined very differently, again, from ordinals and cardinals.
- In cardinals $\alpha^\beta$ is the cardinality of the set of all functions from $\beta$ to $\alpha$.
In the ordinals, we care about the order, so $\alpha^\beta$ is the order type of the reverse lexicographic order of functions from $\beta$ into $\alpha$ which are non-zero only in finitely many coordinates.
Equivalently, and perhaps more clearly, we can define this by induction, $\alpha^0=1$, $\alpha^{\beta+1}=\alpha^\beta\cdot\alpha$, and for a limit ordinal $\delta$, $\alpha^\delta=\sup\{\alpha^\beta\mid\beta<\delta\}$.
Now we easily see that $2^\omega=\omega$ when we are talking about ordinal exponentiation. $2^\omega$ is the limit of $2^n$ for finite $n$, but $2^n$ has a finite order type, and it's a strictly increasing sequence. The limit of a strictly increasing sequence of finite ordinals is $\omega$ itself.
Well, it sounds weird, isn't it? But it's not really weird. We have this sort of phenomenon in other - more familiar - systems of arithmetics.
In the natural numbers $n\cdot m$ can be defined as repeated addition of $n$, $m$ times. Addition itself can be defined as repeated successor operation. On the other hand, when we consider the real numbers $\sqrt2\cdot\sqrt2$ cannot be thought as repeated addition. What does it even mean to add something $\sqrt2$ times? It is true that in this case, if we restrict back to the natural numbers then the operations become repeated application of the "previous operation", but this is because of the nature of the natural numbers as a corner stone of modern mathematics (in many many ways). For infinite, and less-corner stoney this is not the case, as exhibited in ordinals and cardinals.
if two ordinals are equinumerous to $A$ does that not make the ordinals the same since every ordinal is the successor of the previous?
No. Both $\omega$ and $\omega + 1 = \omega \cup \{\omega\}$ are equinumerous to $\omega$, but they are not the same.
if the ordinals equinumerous to $A$ are all the same size then why do we pick the minimum in particular to be the cardinal of $A$
Because it's convenient and always exists. There isn't a second-minimum ordinal equinumerous to $7$, for instance. On the other hand, there isn't a maximum ordinal equinumerous to $\omega$ (all ordinals below $\omega_1$ are equinumerous to it). So what other choice do you propose?
how does this miminum ordinal actually correspond to the size/cardinal of the set $A$?
It's the smallest ordinal that has the same size as $A$. That's it.
Best Answer
In general in mathematics, we're not really concerned with what something is, rather with what can be done with it. So if you want to define real numbers as Dedekind cuts, and I want to define them as equivalence classes of Cauchy sequences of rationals, we don't need to argue about it: there is a one-to-one correspondence between your "real numbers" and my "real numbers", which preserves all algebraic and analytic structures. It's the same with cardinal numbers. Set theorists want to define them as certain sets; maybe you want to define them some other way. It doesn't matter as long as you have a consistent definition which satisfies all the properties that cardinal numbers should have.