Talk:Rayo's number

Encyclopaedic?

I'm no math expert so I'm not in a position to comment on what it should be, but is "His number is the largest number known" correct, and up to the quality standards of an encyclopedia? Wouldn't Rayo's number + 1 be larger and therefore the largest number known number? And therefore Rayo's number * 2, Rayo's number squared, etc? There are an infinite number of known numbers larger than Rayo's number? 69.224.107.100 (talk) 17:04, 30 March 2018 (UTC)

Yes you're definitely not an expert. None of those numbers are computationally interesting or novel. I'm not even a fan of the nonsense about the fast growing hierarchy dominating Rayo(n) when you define the fundamental sequence for omega to be Rayo(n). That also is uninteresting and not in any way novel or contributing anything useful.

It is not even true, that there is "no known upper bound" for Rayo's number (which is Rayo(10¹⁰⁰)). The 8 unilluminating examples of larger numbers in this video are Rayo(10¹⁰⁰)+1, 10^Rayo(10100), Rayo(10¹⁰¹⁰⁰), Rayo(g64), Rayo(Σ(100)), Rayo(TREE(3)), Rayo(Rayo(10¹⁰⁰)) and Rayo¹⁰(10¹⁰⁰).

A more interesting upper bound is FOOT(10¹⁰⁰), where FOOT(n) is the first-order oodle theory. The first-order oodle theory was supposed to generalize nth-order set theory of arbitrarily large n. So, I am pretty sure, that the bound of 10¹⁰⁰ can be substantially reduced, but to be safe, I have used a generous overestimate.

Another upper bound is K(10,000), where K(n) is the K(n) system. K(10,000) is Jonathan Bowers' speculation for little bigeddon. So, K(10,000) is an extremely huge upper bound for Rayo's number, as it has been shown, that little bigeddon ≫ BIG FOOT ≫ Rayo's number and after I took a look at "Growth rate" of FOOT(n), I suspect, that even K(3000) is larger than Rayo's number. — Preceding unsigned comment added by 84.154.64.199 (talk) 08:03, 13 October 2020 (UTC)

Maybe, you should use the K(n) system for Rayo's number in symbols.

It has been shown, that little bigeddon ≫ BIG FOOT ≫ Rayo's number.

And, Jonathan Bowers speculated, that little bigeddon is something like K(10,000).

So, if you use the K(n) system, then you only need a few thousand symbols for Rayo's number. — Preceding unsigned comment added by 84.151.253.225 (talk) 10:41, 27 September 2020 (UTC)

Computable number way beyond Rayo's number?

Is it possible to make a computable number, that is much bigger than Rayo's number? — Preceding unsigned comment added by 84.154.70.190 (talk) 19:59, 17 June 2020 (UTC)

"Computable" is only meaningful for series, not individual numbers. Any positive integer is computable, by adding 1 + 1 + 1 + 1.... as many times as necessary. 2.24.117.98 (talk) 00:25, 24 June 2020 (UTC)

Responding late, but in case anyone has questions about this topic, it is very common.

A uncomputable number is a term widely used in Googology, although I think it is wrong, as it misleads beginners. Typically this term is used to describe numbers generated by uncomputable functions that have received names and are considered non-trivial extensions, examples are Rayo's Number, Busy Beaver Numbers, Fish N. 7, etc.

In the case of uncomputable functions things get more interesting: uncomputable functions are functions that cannot be calculated using any finite computational algorithm in a finite time, even considering unlimited memory.

In other words, consider f(x): if there is a finite algorithm that can calculate an f(x), then f(x) is computable, even if it takes time and memory that is completely unfeasible, as long as it is a finite.

A computable function does not necessarily grow quickly, there are uncomputable functions that do not grow quickly (an example would be 1/Rayo(x)). When we consider fast growing uncomputable functions it is very interesting because if a function f(x) is uncomputable then it grows faster than any computable function. But what does this mean? Does this mean that f(x) is always greater than any computable function? No. What this means is that there is some value "x" that f(x) will always be greater than the other function with the same parameter "x".

For example, which function grows faster: x^2 or x! ?

It is very easy to prove that x! grows faster than x^2, but that doesn't mean x! is always greater than x^2.

Let's calculate x^2:

x | x^2

0 | 0

1 | 1

2 | 4

3 | 9

4 | 16

5 | 25

6 | 36

7 | 49

...

Now let's look at x!:

x | x!

0 | 1

1 | 1

2 | 2

3 | 6

4 | 24

5 | 120

6 | 720

7 | 5040

...

You can see that for x < 4, x^2 > x!

but for x >= 5, x! >x^2

And this is true for all x >= 5, in this case we say that x! "grows faster" than x^2.

The same thing happens with uncomputable functions, it will grow faster than any computable function, but it does not mean that for any uncomputable function f(x) and any computable function g(x) f(x) > g(x ) for all x. JosterFen (talk) 14:43, 21 February 2024 (UTC)

Just a small addition, uncomputable numbers is an well-defined term, but when we talk about Googology it is used, in my opinion, in the wrong way.

If you want to see truly uncomputable numbers, I recommend reading this article or watching this video from the Numberphile channel. JosterFen (talk) 14:38, 23 February 2024 (UTC)

I don't think ANY number is computable, in cases like this I would suspect the limit in computability will be in the amount of available memory to even store the entire thing.

How big is it?

This article should explain how big it is.2003:C2:BF13:8A0:70A8:C478:3293:D254 (talk) 16:10, 30 April 2021 (UTC)

The math speaks for itself, if it's difficult to understand how it is described then it will be much more difficult to understand how to compare it to anything else that would show just how big it is.

What does it start with?

Here are the 10 rightmost decimal digits of Graham's number:2464195387. What are some of the decimal digits of Rayo's number? Thanks 78.144.48.161 (talk) 21:54, 1 May 2024 (UTC)

We don't know anyone JosterFen (talk) 18:48, 17 May 2024 (UTC)