# Thread: Is the Rayo of Graham's number larger than the Rayo of TREE(3)? Could we ever know?

1. ## Is the Rayo of Graham's number larger than the Rayo of TREE(3)? Could we ever know?

I was thinking. Rayo's number, which is meant to be the largest finite number ever concieved, is defined as:

The smallest number bigger than any number that can be named by an expression in the language of first order set-theory with less than a googol (10^100) symbols.

Rayo's number must exist somewhere. Presumably we don't have the symbols to express it yet.

But there must be a method to attain a "provisional Rayo" based on the current symbols we have.

The largest numbers we know of are Graham's number, TREE(3) and SSSGC(3).

Graham's number can be written very simply as G64.

Presumably though, we could extend that notation to higher "powers":

GG
64 etc, a googol times.

Doing so only increases the already small number of digits by one each time.

To do the same to, say, TREE(3) you would have to add five symbols: TREE(TREE(TREE, plus god knows how many closing brackets.

So even though TREE(3) is a much larger number than Graham's number, is Rayo Graham larger than Rayo TREE(3)?
Last edited by parallaxicality; 2021-May-05 at 05:59 PM.

2. Originally Posted by parallaxicality
I was thinking. Rayo's number, which is meant to be the largest finite number ever concieved
That won't last long, there is always Rayo's number plus one!

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Originally Posted by parallaxicality
The largest numbers we know of are Graham's number, TREE(3) and SSSGC(3).
This isn't correct. Graham's number is the 64th number in a certain sequence but there's nothing stopping you from continuing the sequence. If we use the notation of g(64) for Graham's number then there's also g(65) which is bigger, and g(66) and so on. Graham, when conceiving of this sequence, was working on a problem which required the 64th number in that sequence so that's why this particular number became famous, but the sequence doesn't stop there. The same is true for TREE(3), it's famous because it is the point where the TREE sequence suddenly explodes: TREE(1) = 1, TREE(2) = 3, TREE(3) = stupendously large number. But again the sequence doesn't stop there, you can continue on to TREE(4) and TREE(5) and so on. Numberphile had a good video on these particular sequences and their growth rates, see here: https://www.youtube.com/watch?v=0X9DYRLmTNY

As for Rayo's number, note that its definition explicitly says "in the language of first-order set theory" - the italicized bit is important here, as first-order set theory is much less expressive than second-order set theory. I'm not sure if either Graham's or the TREE sequence can even be expressed in first-order set theory.

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Originally Posted by caveman1917
As for Rayo's number, note that its definition explicitly says "in the language of first-order set theory" - the italicized bit is important here, as first-order set theory is much less expressive than second-order set theory. I'm not sure if either Graham's or the TREE sequence can even be expressed in first-order set theory.
Turns out you can express Graham's and the TREE sequence in first-order set theory. Coming back to the original question, given that Rayo(n) is monotonically increasing in n, and TREE(3) > g(64), therefor Rayo(TREE(3)) >= Rayo(g(64)). However, again, these are not the largest known numbers, they are just large numbers that are famous for some reason, but you can keep constructing larger numbers from them if you want. Indeed, Rayo(TREE(g(64)) is going to be even larger.

5. For fans of this I recommend https://googology.wikia.org/wiki/Googology_Wiki the Googology Wiki.

6. Where are they getting those names? I'm assuming drugs are consumed in large quantities on that site.

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