Let $\alpha$ be an ordinal number and define $f_\alpha$ as:
- $f_\alpha(0) = \alpha + 1$
- $f_\alpha(n+1) = \omega^{f_a(n)}$
Let $S(\alpha) = \sup\{f_a(n)\ |\ n \in \omega\}$
Then $S(\alpha)$ is an epsilon number and is the least epsilon number greater than $\alpha$.
Since none of natural numbers are epsilon number, I think $S(n)=S(m)$ for every natural numbers $n,m$. I know that I'm wrong but I don't know why. Please, help.
And I have problem with showing that $m