User blog comment:MilkyWay90/Not-Registered Users, tell your Googology ideas in the comments/@comment-2601:142:2:EC49:247C:A73D:AD4A:5ED9-20180803125948/@comment-30754445-20180804212851

Great. So let's roll with it:

<5,n> ~ [zeta-0]n

<6,n> ~ [gamma-0]n

<7,n> ~ [SVO]n

<8,n> ~ [LVO]n

<9,n> ~ [BHO]n

And now things get tricky and much more subjective. But here is how I will continue:

<10,n> ~ [ψ(Ω2]n (multiple collapsion functions)

<11,n> ~ [ψ(Ωω]n (defining collapsion functions simulatnously)

<12,n> ~ [ψ(ΩΩ]n (indexing collapsing functions with ordinals)

<13,n> ~ using inaccessibles (I)

<14,n> ~ using mahlos (M)

<15,n> ~ using weakling compacts (K)

<16,n> ~ reflection principles (Pi4 reflection)

<17,n> ~ stable ordinals

<18,n> ~ Pi-12-CA (this is a huge jump, basically transitioning from defining numbers with OCF's and defining them by the strength of axiomic systems. But I think it would be more natural to make this jump then to continue increasing the complexity of our OCFs... Then again, this may just be a statement of my own ignorance.

<19,n> ~ 2nd order arithmetic

<20,n> ~ n-th order arithmetic (Loader's Number would around here)

And now, again, we reach a gap in my own understanding. We should probably have the PTO of ZFC somewhere nearby, but I'm not well versed enough in proof theory to tell whether it is reasonable to go directly from n-th order arithemtic to ZFC (my guts tell me it probably isn't). But since this is all wild speculation and good fun, why not continue:

<21,n> ~ 1st order set theory

<22,n> ~ ZF/ZFC (will result in comparable numbers either way)

<23,n> ~ ZF/ZFC + large cardinal axioms

And here, finally, we completely run out of ideas for step-by-step things, so we'll just have to jump to:

<24,n> ~ BB(n) (basically, the largest number which can be computed by an computer program with n letters in a given computer language).

(funny how we got to this point way before <100,n>. Well, you wanted to follow the bold approach, didn't ya? As the saying goes: be careful what you wish for :-) )