User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-5150073-20130414231904/@comment-5529393-20130416142057

I prefer the following definiton for omega_1^{CK}[n]:

Let f(n) be the largest ordinal definable using an n-state Turing machine. By "defining an ordinal using a Turing machine", I mean a Turing machine that defines a well-ordering relation on the natural numbers, that happens to have that ordinal as its order type. Then you just weed out the "bad" values of f(n), i.e. let omega_1^{CK}[n] be the nth ordinal in the range of f(n).

This, in my opinion, is simpler and and less ad hoc than your definition using complexity. The reason I didn't use it above is that I was already using Kleene's O do define the fundamental sequences of all the ordinals up to omega_1^{CK}, so I figured, I might as well use it for omega_1^{CK} as well. As for f_w_1^CK+w_1^CK(n), I think it's best to use oracle Turing machines to define fundamental sequences for all ordinals up to omega_2^{CK} (and above) uniformly.