User blog comment:Vel!/Ultracardinals/@comment-10429372-20140619084656

The first weak ultracardinal is not so big. If my calculations are correct, then the weak ultracardinal is \(\Beth_{\Beth_0}\).

First we see that \(C(\alpha) ≤ \alpha\).

Then we note that, for any \(\Beth_{k}\) for finite k,

\(lim{\Beth_{k}, \Beth_{k}^{\Beth_{k}}, \Beth_{k}^{\Beth_{k}^{\Beth_{k}}}, ...\}\) =

\(lim\{\Beth_{k}, \Beth_{k+1}, \Beth_{k+2}, ...\} = \)

\(\Beth_{\Beth_0}\)