# How Big would “Graham's Tree” be?

What if in Graham’s Number every “3” was replaced by “tree(3)” instead? How big is this number? Greater than Rayo’s number? Greater than every current named number?

• What is "tree(3)"? – Arthur May 21 '17 at 12:26
• googology.wikia.com/wiki/TREE_sequence – Goodwin Lu May 21 '17 at 12:27
• By the way, "every current named number except infinity" makes no sense. Infinity is not a number – Arthur May 21 '17 at 12:31
• ah, alright. Some consider it to be. – Goodwin Lu May 22 '17 at 18:54
• Rayo's number is by design almost certainly larger than anything along these lines you could write down. – Mark S. May 22 '17 at 19:15

## 3 Answers

The TREE function grows much much faster than any construction of knuth up arrows. Because of this, inserting the TREE function into Grahams number would yield a number still very close to TREE(3). It would be like trying to create a number larger than a googolplex by adding a 1 on the end. You would be better off inserting Grahams number into TREE instead of the other way around, creating a "TREE's Graham" instead of "Graham's TREE"

No. If you replaced all the $3$'s in the construction of Graham's number with $\operatorname{TREE}(3)$, the resulting number would be smaller than $g_{\operatorname{TREE(3)}}$ where $g_n$ denotes the $n$th number in Graham's sequence with $g_{64}$ being Graham's number. This is much much smaller than $\operatorname{TREE}(4)$, for example.

• Certainly smaller than $TREE(g_{TREE(3) + 64})$ – Francisco José Letterio Jan 13 '18 at 4:09
• Though $\operatorname{TREE}(4)$ is already a good enough upper bound. – Simply Beautiful Art Jan 13 '18 at 13:25

The numbers themselves are already so big that doing that would barely change it at all. It would still be ZERO compared to a number like Rayo's number. Obviously, if doing so really did make it the largest number ever created, why wouldn't people do it earlier?