What is the least ordinal $\beta$ for which the function $f_\beta(n)$ in fast-growing hierarchy is incomputable? Fast-growing hierarchy consists of a transfinite succession of faster growing functions $f_\alpha$:
$f_0(n) := n+1$,
$f_{\alpha+1}(n) := f^n_\alpha(n)$,
$f_{\alpha}(n) := f_{\alpha[n]}(n)$ if $\alpha$ is a limit ordinal where $\alpha[n]$ denotes the n-th element of the fundamental sequence assigned to the limit ordinal $\alpha$.
Ackermann function grows about as fast as $f_\omega$. Some functions like busy beavers grow faster than any computable function. My question is:
What is the least ordinal $\beta$ for which the function $f_\beta(n)$ in fast-growing hierarchy is incomputable?
Clearly, $\beta$ is no greater than Church-Kleene ordinal $\omega^{CK}_1$ since that is the incomputable ordinal itself, but could $\beta$ be smaller, e.g. $\epsilon_0$?
 A: That depends on the system of fundamental sequences that is used.  If the system of fundamental sequences for all limit ordinals less than or equal to $\beta$ is computable, then $f_\beta$ will be computable.
More precisely, say we have a computable ordinal $\beta$ and some injective function $g: \beta+1 \mapsto \bf{N}$ such that the image of the limit ordinals, the image of the successor ordinals, and the predecessor function $j(g(\alpha+1)) = g(\alpha)$ are all computable.  Suppose further that there is a computable function $h: \bf{N} \times \bf{N} \mapsto \bf{N}$ such that, for each limit ordinal $\alpha \le \beta$,  $g^{-1}(h(g(\alpha),n))$, as a function of $n$, gives the fundamental sequence for $\alpha$.  Then $f_\beta$ is computable.
Note that it would not be sufficient if for each limit ordinal $\alpha \le \beta$ there would be a computable function $h_\alpha$ such that $g^{-1}(h_\alpha(g(n))$ gives the fundamental sequence for $\alpha$.  A counterexample would be to set $\omega (n+1) [m] = \omega n + BB(n+1)+m$ and $\omega^2 [m] = \omega m$, where $BB(n)$ is the busy beaver function.  Then $f_{\omega^2}(n)$ would exceed the busy beaver function.
Also, it is not clear to me that $f_{\omega^{CK}_1}(n)$ must be incomputable.  Certainly we cannot compute it through an ordinal notation for $\omega^{CK}_1$, but it may be computable some other way.
