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I made the horrible mistake of hosting a Scott Aaronson style "large numbers" competition. (I used a character limit to prevent it from getting too crazy.)

I was able to knock out most of the contenders myself, but there are two left for which I'm uncertain, and I could use some help. (Actually three, but I won't worry about the third one here.)

Firstly a notation for function composition: define Comp(f, input, repetitions) to mean applying function "f" to a number "input", then applying f to the result, and repeat until f has been applied this way repetition times. For example, Comp(f, 2, 3) = f(f(f(2))).

The two contenders (simplified a bit from how they were entered) are:

  • Rayo(Comp(BB, G64, G64 + 2))^2
  • Comp(TREE, 9, TREE(G64))^2

Yes these are both salad numbers. (The entrants were not very sophisticated.) I'd like to determine which is larger.

I'm aware that proving statements about such large numbers is extremely difficult (especially when they include uncomputable functions like the first one does), so I'd accept a strong heuristic argument instead.

I'm inclined to think that the first is larger, since Rayo() and BB() both grow faster than any computable function like TREE(). But TREE() starts out larger for small values, and we don't know where the other functions finally cross it, so this isn't conclusive.

Isaac King
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1 Answers1

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Rayo's number $=\text{Rayo}(10^{100})$ is generally considered one of the largest defined and well-known numbers, and while I don't know of any proofs of this, there are a few results, such as $\text{Rayo}(7901)>\text{BB}(10^{100})$ and $\text{BB}(18)>$ Graham's number, which strongly suggest that it is indeed as large as is suspected. This means that $$\text{Rayo}(\text{Comp}(\text{BB}, G_{64}, G_{64} + 2))^2>>\text{Rayo}(10^{100})\stackrel{?}{>>}f_{\psi(\Omega_w)}(2)>>f_2(f_{\psi(\Omega^{\Omega^\Omega})+1}(f_{\psi(\Omega^{\Omega^\Omega})}(f_{\omega+2}(64))))>f_2(f_{\psi(\Omega^{\Omega^\Omega})+1}(f_{\psi(\Omega^{\Omega^\Omega})}(G_{64})))=f_2(f^{f_{\psi(\Omega^{\Omega^\Omega})}(G_{64})}_{\psi(\Omega^{\Omega^\Omega})}(f_{\psi(\Omega^{\Omega^\Omega})}(G_{64})))>f_2(f^{f_{\psi(\Omega^{\Omega^\Omega})}(G_{64})}_{\psi(\Omega^{\Omega^\Omega})}(9))\stackrel{?}{\gtrapprox} \text{Comp}(\text{TREE},9,\text{TREE}(G_{64}))^2$$ where $f^a_b(n)$ denotes the fast-growing hierarchy and $\psi(a)$ refers to Buchholz's ordinal collapsing function in which $\psi(\Omega^{\Omega^\Omega})$ is known as the large Veblen ordinal (LVO). The main problem with this argument is the last step, as lower bounds for TREE below and near the level of the LVO are known, but no explicit upper bounds are known, making an accurate comparison infeasable at this time. However, the best known lower bounds for TREE are very small compared to the largest well known computable functions, like $f_{\psi(\Omega_w)}(n)$ used above, so it seems highly unlikely that $\text{Comp}(\text{TREE},9,\text{TREE}(G_{64}))^2$ comes anywhere near $\text{Rayo}(\text{Comp}(\text{BB}, G_{64}, G_{64} + 2))^2$.

David Chew
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  • Thanks! I believe there's an upper bound for TREE provided here: https://math.stackexchange.com/questions/1950116/where-does-treen-sit-on-the-fast-growing-hierarchy/1959090#1959090 – Isaac King Jun 25 '23 at 17:31
  • After reading that answer and the comments, it looks like no proof has actually been written yet, though it does seem likely that the upper bound should be close to the lower bound. – David Chew Jul 01 '23 at 03:12