User blog comment:Kolmogrov142 Oracle Tuto X/Building up the fast growing hiearchy/@comment-31663462-20180125044346

Also, you seem to believe that $$\omega[n]=n$$ must be true. But note that $$\omega$$ is also the limit of the natural numbers squared, so it's not illegal to say $$\omega[n]=n^2$$. In fact, there are infinitely many possible ways for us to have $$\alpha[n]$$ to be defined. This is important because for larger and larger ordinals, there exist multiple often used and recognized fundamental sequences, which may not be equivalent, and that makin large ordinals is half the story. Defining their fundamental sequences is the other half.

Also, Cantor's diagonal argument has nothing to do here. It's about the cardinality of sets and bijection, which is a completely different thing.