Ordinal Arithmetic: Why is ε0 = ω^ω^ω^ω...

Question

In other words, in this notation, why not continue the pattern of ω+ω, ω*ω, ω^ω, ω↑↑ω, and so on, using ω↑↑ω instead of ε0? Is there any reason for ending that pattern with exponentiation, rather than using tetration and other super-operators? Does ω↑↑ω even have the same meaning as ε0?

Answer 1

The principal reason is that the $\epsilon_0$ notation can be used to extend much, much further, because Knuth up-arrow notation handles infinities poorly. One way of writing $\epsilon_1$ is as $\epsilon_0\uparrow\uparrow\omega$, and then $\epsilon_2 = (\epsilon_0\uparrow\uparrow\omega)\uparrow\uparrow\omega$, and so on. I think we could define $\epsilon_{\omega}$ as $\omega\uparrow\uparrow\uparrow\omega$ (but don't quote me on that one).

But what about $\epsilon_{\epsilon_0}$? That would take $\epsilon_0$-many up-arrows. And then $\epsilon_{\epsilon_{\epsilon_0}}$ is even worse. So, to summarize: we could use Knuth up-arrow notation, but we had to start using the epsilon notation somewhere, so we started here.