User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-1605058-20130318105049

I was aware that there is possibility to create such fundamental sequences, but they are non-recursive. There are some reasons for which Kleene's \(\mathcal{O} \) isn't always considered ordinal notation.

There exists extension of this system - while 3*5^m is used for sequence computed by m-th Turing machine, we can use 7*11^m for sequences by 1-Turing machines, we can use 13*17^m for  sequences by 2-Turing machines etc. We can use infinitude of primes to reach \(\omega ^{CK}_\omega \)

We can do even better - we can well order primes in such way: 3<7<13<...<5<11<17..., and if we take consecutive pairs from this ordering we can reach \(\omega ^{CK}_{\omega 2}\) and so on. With ordering with order type \(\alpha \) we can reach \(\omega ^{CK}_\alpha \), so this already is strong system