Is the ordinal $\omega \uparrow^\omega \omega$ still recursive?

Question

In this question, a very large countable ordinal $\omega \uparrow^\omega \omega$ is defined. Is this ordinal still recursive?

Answer 1

Yes. First, by induction, for each $n \in \omega$, $\omega \uparrow^n \omega$ is computable, uniformly in $n$. Second, $\omega \uparrow^\omega \omega$ is just the limit, which is to say the sum, of these.