Cantor's diagonalization argument shows that the power set of natural numbers is larger than $\aleph_0$, that is, it has a larger cardinality. Every natural number could either be in any given set of the superset, or it could be out of said set. So, that gives two possibilities for every natural number. Since there are infinite natural numbers, that is $2^\omega$ different sets. Therefore, the superset of the natural numbers has an order of $2^\omega$. Therefore, $2^\omega$ has a larger cardinality than $\omega$ because it can not be counted, as it is equal to the superset of natural numbers, which is of a larger cardinality then the set of all natural numbers.
If this were a valid proof, then $\omega_1$ would not be the first ordinal that is higher than $\omega$, as is the current mathematical consensus, as far as I know. Why is this not a valid proof, or what have I misunderstood about ordinals and cardinals?