User blog comment:Tetramur/Pentational arrays and beyond - comparisons/@comment-37993808-20191211162933/@comment-37993808-20191212105353

Ok.

1. The ordinary Busy Beaver function is proven to be of order at least \(\omega_1^\text{CK}\)  in FGH, and many googologists tend to believe that it is actually the order of growth (but ordinal is still computable, because there are only countably many Turing machines).

2. The Busy Beaver function with oracles (order 2, 3, 4, ..., \(\alpha)) has the order of growth of at least \(\omega_{alpha}^\text{CK}\) in FGH.

3. Fish fourth function continues the idea of an oracle. So it is of order \(\omega_{alpha}^\text{CK}\), where \(\alpha) is ordinal for F3 in ordinary FGH.

4. Adam Goucher, the inventor of xi function, gave us some evidence of order of it. It is probably the first fixed point of (\alpha →\omega_{alpha}^\text{CK}\), but it is still a computable ordinal, as there are only countably many combinators.

5. Doodle function - no clues. It is still a computable ordinal, as there are only countably many combinators.

6. Rayo's function - Fish proved that there is no way to write it compactly inside of FOST and, consequently, it eventually dominates all functions that are defined compactly in FOST or weaker theories. So, I think that the ordinal representing Rayo's function is uncomputable, and in fact, it is possibly the smallest uncomputable ordinal.

7. Fish seventh function is a continuation of Rayo's function and if latter is strongly uncomputable then so is F7.