User blog comment:ArtismScrub/Ordinal factorial notation?/@comment-5529393-20170922201650

Some suggestions:

Your definitions skip ordinals - for example, you say what happens at n!(ωm), but not what happens at n!(ω^2+ωm). Going by your examples one would expect this to be n!(ω^2+ω(m-1)+n), but it's less clear further on down the line; I can't tell what we should do with n!(ω^(ω^3+ω^2)), for instance. To take care of all cases, it helps to have more general rules. For example, you give a rule for n!(Y)+1 for all ordinals Y (that should actually be n!(Y+1) though); you could also give rules for n!(Y+ω) when Y is a multiple of ω, or for n!(Y+ω^m) when Y is a multiple of ω^m, and that would handle much larger classes of ordinals.

Also, take care that your rules for limit ordinals actually give you larger numbers than smaller ordinals, assuming this is a feature that you want. For example, n!(ω^ω^2) = n!(ω^n!ω) looks dangerous - it reduces ω^ω^2 immediately down to ω^n!ω, which is ω to a finite power, so it's not clear that it's going to be faster growing than n!(ω^(ωa+b)), for finite a and b, which goes through many more reductions. Although now that I think about it, it probably is; I'm just saying that usually, we would reduce ω^ω^2 to ω^(ωn), so that it would be almost automatic that the function at ω^ω^2 would grow faster than any ω^(ωm) and therefore any smaller ordinal. Of course, you may not want to follow what other people do.

I don't know if you care, but your notation is not all that strong. For example, n!(ε0) = n!(ω↑↑n) is less than the function n!(ω^n) applied n times starting from n, whereas for the fast-growing hierarchy, f_{ω^ω+1}(n) is already f_{ω^ω} applied n times. This is largely an artifact of reducing your ordinals greatly in your rules, for example n!(ω↑↑n) becomes n!(ω^n!(ω↑↑(n-1))), which is ω to a finite power. So if you are looking for something comparable to the fast-growing hierarchy, you need stronger rules.