User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-5150073-20130414231904/@comment-5150073-20130417123650

For fundamental sequence to w_2^CK, we can look at the fact that each recursive ordinal can be expressed in terms of hyper-operators and w. For example, e_0 = w^^w, z_0 = w^^^w, n_0 = w^^^^w, and so on. We can say that between w_1^CK and w_2^CK there exist analogues of e_0, z_0, n_0, etc. These analogues just used w_1^CK instead of w. For example, if w_1^CK[n] = e_0 = w^^w, then w_2^CK[n] = (w_1^CK)^^(w_1^CK). That also applies to the higher admissible ordinals, with our n: w_10^CK[n] = (w_9^CK)^^(w_9^CK).