At each step the partial summand is about ln(N), so about n, which has about ln(n)/ln(10) digits (left to the decimal point). Since you multiply by 10^N, you get about N+ln(n)/ln(10) digits, so about e^n+ln(n)/ln(10) digits. Since you do that G times, the total number of digits is approximately the sum of e^n+ln(n)/ln(10) for n from 1 to G.
Each term is equivalent to e^n, so this is about sum(e^n) for n from 1 to G, so about eG+1/(e-1).
1 Like
It seems I indeed still do not understand Paris-Harrington, then 
That’s a nice trick actually with the geometric series for big enough numbers 
Googologists call this kind of number a “Soup number”, because you’re combining an interesting construction (Rayo’s number) with some lesser ingredients that don’t move the needle at all on a googological scale.
2 Likes
Never even heard of that word, but there’s a wiki and a Reddit XD Googology.
Very fun!
Are you a googologist?
Yes, you could call me that. I co-wrote a paper called
“A Googolplex of Go games” [1]. And proposed/analyzed a new busy beaver function [2].
[1] https://matthieuw.github.io/go-games-number/AGoogolplexOfGoGames.pdf
[2] A333479 - OEIS
3 Likes
Is there a description of this scale?
Can you prove
ggg < Rayos number
?
I might, if I knew what g_g_g was?!
Basically, if you can define it in less than 10^100 symbols of first order set theory, then it is by definition less than Rayo’s number.
1 Like
With the requirement to find it mentioned in the public “literature” …
Perhaps, this is rather a contest for Googleogists
3 Likes
It’s:
1 Like
If no one can prove that their number is bigger, then no one else can win.
My submission might seem silly, but a lot of the other large numbers reduce to something like it. For sufficiently large n, we can’t find a finite upper bound on BB(n): make a Turing machine with n states to search for inconsistencies in ZFC. Then run it for at least BB(n) steps, and you get a proof (or disproof!) of consistency of ZFC. Thus any number we can in principle write down after a finite computation cannot provably beat something like BB(10^100). I think the same thing holds for Rayo’s number.
XD
Although we do have
1 Like
If you were the first to submit your number, ok. But since 1000 was submitted first, and you can’t prove your number is larger than 7…
The googological way to compare two numbers is in the Fast Growing Hierarchy (FGH) [1]. Although this is a hierarchy of functions, not numbers, we can tie a number to the lowest function f in the hierarchy for which the number can be expressed with a smallish input.
In the case of Graham’s number, that function is f_{ω+1}, since G64 is about f_{ω+1}(64). We cannot reasonably express G64 as f_ω(n), since by definition f_{ω+1}(64) = f_ω^64(64) = f_ω(f_ω^63(64)), and n=f_ω^63(64) is still huge.
So we can say that G64 “sits at” ordinal ω+1.
The number PH(9), courtesy of Paris & Harrington, sits at ordinal ε0 = ω^ω^ω^… [2], which obviously dwarves ω+1.
The class of ordinals used to index functions in the FGH is indescribably big, and ε0 is just getting their feet off the ground.
I highly recommend this video series [3] to get a taste of how incredibly rich the class of ordinals just up to Buchholz Ordinal (which is where the so-called Slow Growing Hierarchy catches up to the FGH) is.
Btw, the busy beaver function is believed to correspond to f_ω1CK, where ω1CK is the Church-Kleene ordinal, which is the limit of all recursive ordinals. It also separates all the “computable” large numbers, as well as all busy beaver numbers, from “uncomputable” numbers like Rayo’s, with the latter sitting at PTO(ZFC), the Proof Theoretic Ordinal [4] of Zermelo Fraenkel Set Theory with the axiom of Choice.
[1] Fast-growing hierarchy - Wikipedia
[2] Epsilon number - Wikipedia
[3] https://www.youtube.com/playlist?app=desktop&list=PL3A50BB9C34AB36B3
3 Likes
Btw, here are two very recent videos about the largest number you can write down:
3 Likes
One thing I am curious about, what does Rayo’s sequence look like?
Rayo(1), Rayo(2), … or whenever it begins to be defined.
What’s my minimal amount of symbols I need just to set up the Natural Numbers in first order set theory?
Or is it that I have my bunch of axioms already laid out, the natural numbers exist etc.
How many symbols does the number 10 require in first order set theory in order to use it in order to describe some other number, or the number 10^10?
Do I need to write down S(S(S…(1)…) for successor functions or?
Edit: I guess someone made a video, I’ll watch that
1 Like
My Googleogist skills help me find this article in the Googology wiki: Rayo's number | Googology Wiki | Fandom
It’s very difficult to compute the number exactly, but it appears that people have worked out the exact values for smaller numbers. It seems that the sequence starts quite slowly
I guess since it takes a certain amount of symbols just to start stating anything meaningful. However, after 320 symbols, it seems to blow up quite rapidly
However, these are lower-bounds, so it may be possible to discover constructions that yield larger numbers.
2 Likes
I’m guessing there’s some minimal number of symbols you need to define some normal (more familiar) fast growing functions like exponentials, factorials and so on, and then maybe once you have those, iterating composing and so on probably makes the numbers you can achieve grow rapidly then.
1 Like