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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?

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

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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.

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

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