Ordinals and Cardinals larger than the fixed points of $α↦ω_α$ and $α↦\aleph_α$?

Question

Surely there is no limit to how high we can go, so how do we talk about ordinals and cardinals higher than the fixed points of the functions $α↦ω_α$ and $α↦\aleph_α$? Is the power set of the $\aleph$-fixed point even larger than the $\aleph$-fixed point itself? Or is the $\aleph$-fixed point so mind-bogglingly huge that it can't be rivaled by the same methods we have previously used to reach large cardinals?

Answer 1

Cantor's theorem says that $\mathcal{P}(x)$ is always strictly greater than $x$; it doesn't matter whether $x$ is an $\aleph$-fixed point or not. Similarly, every cardinal has a successor cardinal which can be constructed just as we construct $\omega_1$ from $\omega$.

Note that any strictly increasing operation on cardinals, like the successor operation $\kappa\mapsto\kappa^+$, can get us out of the "fixed point trap." For example, there is no fixed point of the map $\alpha\mapsto\omega_\alpha^+$. So we can quite easily go well beyond the least $\aleph$-fixed point.