User blog comment:Simplicityaboveall/Insanely Fast-Growing Functions/@comment-25912386-20171111063151/@comment-30754445-20171112000330

Joe, I enthusiastically share your philosophy of keeping things as simple as possible.

And you're right that, unfortunately, many googologists don't really care for simplicity. Some of them actually enjoy the craziness of doing things the complicated way, and others simply aren't interested in putting the effort to explain what they're doing in clear and simple terms.

But the truth is, 90% of what the people on this website are doing is actually quite simple to understand if it is explained properly. One does not need to know set theory, in order to understand how ordinals or "the fast growing hierarchy" works. The concept, at its core, is actually very simple. In fact, the entire concept of "ordinals" can be summarize in one  sentence:

"An ordinal represents a structure in which items are placed in a given order"

So any finite number is an ordinal. For example, the number "5" can represent a row of 5 apples.

So what does ω represent? It's a row of apples of arbitrary length. You can think of it either as "an infinite row", or as a row that just has the potential of going as far as you want it to go. The point is that if you have an "ω structure" of apples, then you can fetch the nth apple in that row, for any arbitrary large n.

Then we have ω+1. That's one row of apples, plus a single apple that comes after that entire row. Something like this:





ω+2 would, of course, be this:





Then we have ω×2, which is simply two full rows of apples:





Note that all the apples in the second row come after the first row. The order is important (this is why these things are called ordinals. They tell us the order in which the items are placed)

Then we have ω×3 as three rows. and so on. For example, here is a representation of the ordinal ω×5+3:













(that's 5 rows of potential infinity, plus 3 apples in the end)

Now we can also have an arbitrary number of rows, which leads us to ω2 = ω×ω:













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Once again, you are free to choose between visualizing this as a grid that actually goes on forever in both dimensions, or as a finite square that can dynamically grow to any arbitrary size. Either way, we can fetch the nth apple in the mth row from this grid of apples, no matter how large n and m are.

We can of course have more grids, which eventually leads us to ω3. This can be viewed as a cube of apples stretching to (potential) infinity.

For higher powers of ω we'll need a different analogy, because we can't visualize more than 3 dimensions (there are intuitive analogies that go at least up to ε₀. I'll be happy to go into that in a seperate post if you wish). At any rate, for now, let us stay at the ω2 level and dig a bit deeper:

How does the FGH look at this level? Well, basically, the functions in the FGH up to any given ordinal, can be placed in the structure that is represented by that ordinal. As we've already seen, ω2 can be represented an ordinary 2D grid, so we have the following order:

f0, f1, f2, f3, ...

fω, fω+1, fω+2, fω+3, ...

fω×2, fω×2+1, fω×2+2, fω×2+3, ...

fω×3, fω×3+1, fω×3+2, fω×3+3, ...

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And the actual values of any function in this grid can be calculated by the following rules:

(i) f0(n) = n+1.

(ii) If the function in question isn't the first in its row, then we iterate the function immediately to the left n times.

For example: f7(100) = f6100(100)

(iii) if the function is the first in its row, then we evaluate the nth function on the previous row for the same n:

For example: fω×2(100) = fω+100(100)  = fω+99100(100) = ...

See? It's actually quite simple. And this grid of functions is already far more powerful than the current version of your own functions. In fact, every time you did the "h to H" or "g to G" thing, you went exactly one row down the grid. Since you did it exactly twice, your final function is comparable to the functions on the 3rd row.

Now, this basic concept remains unchanged for all larger ordinals. This is how the entire FGH is defined.

There is, however, a catch:

As we go higher and higher, the structures become more complex. This is exactly why we need ordinals in the first place: Because if you want to create a system that does a certain level of recursion, you must master the recursion structure in question.

The good news is that most of the hard work was done for us. The best mathematical minds of the 19th/20th century spent decades of work researching this topic of "ordered structures" (which are also known as ordinals). It is much much easier to learn what they have done, than it is to try and invent the same structures on your own.

In short: Understanding googology (of computable functions, at least) cannot be seperated from understanding ordinals. So if your goal is to create really fast growing functions that a layman can understand, then you should:

(1) Learn in depth how ordinals work up to some decent level (ωω seems like a good starting point).

(2) Look for ways to explain and/or visualize these ordinal structures in layman-friendly language.



Incidently, there are simple notations which manage to do an insane amount of recursion. The catch is that while using these notations can be easy, actually understanding how they work (and why they grow so fast) is a completely different matter.

For example, this seemingly innocent notation:

http://googology.wikia.com/wiki/Pair_sequence_number

Manages to "pack" a completely insane recursion structure which is known as the ordinal "ψ(Ωω)". How come it grows so fast''? What does this kind of growth rate even mean ''in context? Any attempt at an answer will require, first, an explanation of how the ordinal ψ(Ωω) is structured.