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-30754445-20180804190741

@Syst3ms

"Oh yeah, my bad, he didn't properly define it. My brain just autocompleted the rules."

I get that.

What I don't understand, is why you thought that phi(w,w) is reasonable extrapolation. Why not something lower... or higher?

The way I see it, there's no real pattern to the first four levels. Level 0 is w, level 1 is w*2, level 2 is w^2, level 3 is w^w and level 4 is e0. I don't see any specific pattern here that can be extrapolated.

The only pattern I do see, is that each level introduces a brand new concept that "breaks the mold" of the previous level. We can write this as the following intuitive non-formal rule:

"The (k+1)th level of the system adds the simplest possible new idea, while still making the previous level substantially more powerful"

So for level 5, really, we have three natural choices: epsilon_1, epsilon_w or zeta_0.

Level 6 could be anywhere between epsilon_w and phi(w,0) (or possibly gamma_0)

How fast it progresses, really, would depend on how we extrapolate the concept of "a simple idea". For example, should we count epsilon_(zeta_0+1) as a milestone? It's a point where many people get confused, so perhaps it should be considered as a "new idea" (even though, formally, nothing new happens at this point). OTOH we might take the bold route, and jump directly from epsilon_0 to zeta_0 and then to gamma_0.

Funnily enough, our precise choices won't matter in the long run, because even the most conservative choice of "new ideas" will slowly build up to a huge ordinal at some point. We'll run out of ideas way before we'll reach <10000000,0> (for example).

So, what would be the actual the limit of such an hypothetical notation? This would depend of what kind of ideas we're allowed to add at every step. Let's examine two reasonable possibilities:

(1) We are allowed do add anything we want. In this case, the system will obviously be limitless. Unfortunately, it is possible to show that such a notation is logically impossible to define in its entirety.

(This actually makes sense intuitively: we need every step to be fundamentally more powerful than the previous steps, and we have an infinity of them. There's no way to "mechanize" this process into a single definition, and this intuitive truth can be proven mathematically as well)

(2) We are only allowed to add ideas which can be implemented in a step-by-step fashion. You might think this, too, would be limitless, but it turns out that it isn't. There are actually functions out there which grow faster than any step-by-step system: Faster than Veblen functions, faster than all kinds of OCFs, faster than anything we can theoretically construct in this manner. The limit of all the constructable systems is called "the church-kleene ordinal" (written as ω1ck), which can be thought as "the smallest ordinal which cannot be built by any step-by-step process".

Oh, and one more thing:

If you're wondering how we circumnavigate this barrier when we were "allowed to do whatever we want", the answer is this: at some point, we'll just say something like "and now we'll define <100,n> to be to the largest number which can be generated by a step-by-step system with n symbols". (a word of caution - this general idea would need to be heavily polished before it can actually be used to create an actual function that outputs actual numbers).

Now, when we begin our journey towards more and more powerful functions,  this would seem like a convoluted and unnatural step... not to mention a complete overkill. But as we progress, the usual step-by-step definitions would become harder and harder to defend as "the simplest most intuitive next step that makes our function more powerful". When the choices for the next level are "an OCF that collapses stable++ ordinals and requires 30 pages of  equations to define" and "we'll just take the largest number definable with n symbols", the latter will be way more natural than the former.

(I'm simplifying here. In reality, the largest "step-by-step" numbers won't be created with OCFs at all. There are stronger tools for creating such numbers, but my comment is already way too long so I'll stop here.)