How many omegas are there in $\large f_{\varepsilon_0}$?

Question

For a description look at fast growing hierarchy at wikipedia. $\large f_{\varepsilon_0}$ is not defined any more, it is a power tower of omegas, but how many omegas ?

I found a defition

$$\large f_{\varepsilon_0}(n) = \large f_{{\omega}^{...\omega}}(n)$$ with $n-1$ omega's

but it would be more natural to take $n$ omegas.

Which of these definitions is standard ?

If it would be the first one, $\large f_{\varepsilon_0}(3)$ would "only" be $\large f_{\omega^{\omega}}(3)$.

Answer 1

It largely depends on the definition of the fundamental sequence.

Almost all definitions use $\varepsilon_0[n+1]=\omega^{\varepsilon_0[n]}$, however, there are two conventions for $\varepsilon_0[0]$:

• $\varepsilon_0[0]=0$, [Gallier 1991]
• $\varepsilon_0[0]=1$, [Prӧmel, et al., 1991]

This is because it is natural to think as $\varepsilon_0[0]$ as a tower of zero $\omega$s and thus of $\varepsilon_0[n]$ as a tower of $n$ $\omega$s, but on the other hand, form one you can still go one step down, since $\omega^0=1$.