Since my last question here about the Alephs was too imprecise and thus went over like a lead balloon, I am trying a new and simpler question which asks what I probably should have asked before. 

Under "Beth Numbers" in Wikipedia I read:

"In ZF, for any cardinals $\kappa$ and $\mu$, there is an ordinal $\alpha$ such that:

$\kappa \leq \beth\_\alpha(\mu)$."

But under "Inaccessible Cardinals" I read:

"a cardinal $\kappa$ is strongly inaccessible if it is uncountable, it is not a sum of fewer than $\kappa$ cardinals that are less than $\kappa$, and $\alpha < \kappa$ implies $2^\alpha < \kappa$."

These two passages are troubling to me since they seem to be contradictory. The first one seems to imply that for ANY cardinal one can always find a Beth number which exceeds it. While the second one clearly seems to imply that the first inaccesible and any cardinal larger than it, of which there are uncountably many, of course, are all vastly larger than any Beth cardinal generated by even $\omega$ applications of the Power Set operation could ever be. 

I assume that I am simply missing something important here, and that both statements from the Wikipedia are actually true. But what exactly am I missing??