User blog comment:Mh314159/Natural number recursion -- first 6 rule sets/@comment-39541634-20191031150710

"I think I understood everything before Veblen pretty well, but only some of what came after, would need to spend more time on those."

One step at a time.

I'm telling you this from experience. Don't try to jump too quickly. For now, epsilon numbers are more than enough.

"My original goal was to see how large a number I could define and I don't really want to do it by using pre-existing work."

Mathematics is a cooperative field of study. You should not be afraid to build upon the works of others, as long as you are also adding something unique and fresh and new.

Besides, it's unavoidable. Every time you're using the trick of nested recursions, you are using pre-existing work. Using ordinals is exactly the same thing, just on a higher scale.

I will also add that there's a pretty hard limit to what we can do without ordinals. Even the best googologists will have trouble going beyond epsilon-0 without ordinals. It is not a coincidence that every single power notation we have (Bird Arrays, SAN, HAN, Pair Sequences etc.) was created with ordinals in mind. There's simply no other way to do it.

"Anyway, it doesn't help me with my natural number recursions because I cannot find the clear conversions between the FGH and my recursions."

Yet.

There are three reasons why you find this difficult at this time:

(1) You're current notation is needlessly complex. It's a mish-mash of too many ideas which makes proper analysis difficult.

(2) You may understand the rules of the FGH up to (say) epsilons or zetas, but you have no experience actually using ordinals in an actual situation.

(3) You have zero experience with analyzing notations. It's a not a difficult skill to learn, but - like everything else with googology - you need to take it one step at time.

I suggest you start with something smaller and simpler then what you have now, so you'll get a better understanding of what - exactly - is going on. I know this might seem like a silly waste of time at first, but trust me on this one. Make something simple, and ask questions about it until you are 100% clear about how everything works. Once you lay these foundations, you'll see that progressing further becomes infinitely easier.