6
$\begingroup$

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

  • 4
    Oh, that's true. You know that, for any $\alpha$, S(\alpha)>\alpha and $S(\alpha)$ is epsilon. So, you can enumerate the epsilons in increasing order: $\epsilon_0=S(0)$. Given $\epsilon_\alpha$, let $\epsilon_{\alpha+1}=S(\epsilon_\alpha)$. And for limit $\lambda$, let \epsilon_\lambda=\sup\{\epsilon_\alpha\mid\alpha<\lambda\}. You need to check: 1. Every $\epsilon_\alpha$ is an epsilon. This is clear except if $\alpha$ is limit. 2. Every epsilon is an $\epsilon_\alpha$. This can be proved by considering a least putative counterexample.2012-06-04

1 Answers 1

1

Since the question was answered in comments and we don't like leaving questions unanswered, I'm adding azarel's comment as a CW answer:

Actually, $S(n)=S(m)$ for all natural numbers $n,m$. Since $\epsilon_0$ is the minimal epsilon number above of them.