User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-25418284-20130325203802/@comment-5529393-20130326053535

Are we allowed to use ordinals? If so, then the answer is yes. We simply need to define fundamental sequences for limit ordinals up to \(\Phi(1,0,0)\). For limit ordinals \(\alpha\) not of the form \(\omega_{\beta}^{CK}\), we set up a notation using rank-\(\beta\) Turing machines for the appropriate alpha, and define the fundamental sequence using the smallest notation for \(\alpha\). For ordinals  \(\alpha\) of the form \(\omega_{\beta+1}^{CK}\), define the fundamental sequence as I defined above for \(\omega_1^{CK}\), but use rank-\(\beta\) Turing machines rather than regular Turing machines. And for ordinals \(\alpha\) of the form  \(\omega_{\beta}^{CK}\) for limit \(\beta\) but not of the form \(\Phi(1, beta)\), define \(\alpha[n]\) = \(\omega_{\beta[n]}^{CK}\).

That defines fundamental sequences for all ordinals not of the form \(\Phi(1, beta)\). For ordinals of this form, we simply define fundamental sequences in the same fashion as how we defined them for \(\phi(1, beta)\).

To be precise:

When \(\beta\) is limit, \(\Phi(\alpha, \beta) [n] = \Phi(\alpha, \beta[n]) \).

\Phi(\alpha+1, 0) [0] = 1\);

\Phi(\alpha+1, 0) [n+1] = \Phi(\alpha, \Phi(\alpha+1, 0) [n])\).

\Phi(\alpha+1, \beta+1) [0] = \Phi(\alpha+1, \beta) + 1;

\Phi(\alpha+1, \beta+1) [n+1] = \Phi(\alpha, \Phi(\alpha +1, \beta + 1) [n]) \).

When \(\alpha\) is limit:

\Phi(\alpha, 0) [n] = \Phi(\alpha[n], 0) \).

\Phi(\alpha, \beta+1) [n] = \Phi(\alpha[n], \Phi(\alpha, \beta) + 1) \).

\Phi(1, 0, 0) [0] = 1;

\Phi(1, 0, 0) [n+1] = \Phi(\Phi(1, 0, 0) [n], 0).

So that defines fundamental sequences for all ordinals up to \(\Phi(1, 0, 0)\), which we can then plug into your favorite ordinal hierarchy. If you don't want to use the FGH, you can use Busy Beavers;  BB_{\alpha}(n) is the largest number output by an n-state rank-\(\alpha\) Turing machine.

Obviously we can continue this up through the Extended Veblen notaiton, the Bachmann-Howard notatoin, and beyond, all relativized to the function \(\alpha \mapsto  \(\omega_{\alpha}^{CK}\).