User blog comment:Simplicityaboveall/Extremely Large Numbers 2/@comment-5529393-20160801203256

1 - The googologisms on the list were created by various authors, the most prolific of which is Sbiis Saibian, who has created more than 16,000 googologisms. But there are others, such as Jonathan Bowers, Aarex, cookiefonster etc. The googology article itself was also a collaborative effort of a bunch of different people.

2 - It's supposed to be in ascending order, but there are problems, particularly with the numbers created by Bowers. Basically the consensus here is that Bowers's notation is not well-defined past tetrational arrays, so that we can't really talk about how big his larger numbers are, since currently they are not even numbers.

3 - As I said above, the Bowers numbers are not well-defined. If I read Hollom's notation correctly, the BIGG should be around F_{psi(psi_I(0))}(200).

4-5-6 - The SCG(n) function is known to have growth rate between F_{psi_0(Omega_omega)}(n) and F_{psi_0(epsilon{Omega_omega+1})}(n). So SCG(13) should be in that general range. I've already talked about the Bowers numbers and BIGG. Loader's number is tricky, since the Calculus of Constructions is a very powerful system. Mathematicians have been making ordinal analyses on formal theories for a while now, but they haven't been able to get anywhere near second order number theory, much less higher order number theory, which the Calculus of Constructions is stronger than. It appears that using strong theories to define large countable ordinals and fast growing functions gets you much farther than trying to define them "manually", step by step. So it looks like the ordinal for the Calculus of Constructions (and therefore Loader's number) is currently beyond any recursive ordinal notation that we can come up with. (although Taranovsky believes his notation perhaps reaches this far)

7 - It is not true that an uncomputable function has to be particularly fast growing. For example, you can let f(n) be 1 if the nth Turing machine halts, and 0 if the nth Turing machine doesn't halt. This is an uncomputable function, but it can only be 0 or 1!

However, the "uncomputable numbers" on the list of googologisms do indeed come from functions that grow faster than any computable function. For example, the Busy Beaver function is defined as the largest number of 1's in the output of any halting Turing machine with n states. Let's say we have a computable function f. Since it is computable, we can implement it with a Turing machine, say one with k states. We can also implement g(n) = 2n with say m states. We can use n states to write n 1's on the tape, then use m states to double the number of 1's, then use k states to apply f to it. So BB(n+m+k) >= f(2n), so the Busy Beaver function grows faster. The other functions on the list are more powerful than the Busy Beaver function, so the same proof works for those other functions as well.

8 - I don't think Rayo(1) or Rayo(2) are even defined, since you need a certain number of characters to uniquely define a number in the given microlanguage.