TREE(3) Makes Graham's Number Look Tiny

Added: 2026-05-27

[00:00:00]Give a sense of how monstrously large the number tree three is.

[00:00:06]Okay, well, I think the the best way to talk about tree three and there's a technical definition that requires some expertise in a subject called recursion theory. But one way I I look at it is this. Tree of three is so big that you can't even prove it exists.

[00:00:27]Uh that you can't prove that it exists even in a system like strong system like piano arithmetic. If that were true, that would indeed be mind-blowing because any finite number can be shown to exist depending on your definition of exists. I'm sorry. I'm sorry. I'm sorry.

[00:00:47]>> [laughter] >> I misstated this.

[00:00:49]You can't prove it exists without using two to the one trillion pieces of paper.

[00:00:56]That doesn't mean that it's bigger than two to the trillion. It's incredibly bigger than that. I'm saying even with two trillion pieces of paper that makes sense.

[00:01:07]You can't possi- you're not even proving it exists. Right.

[00:01:11]I don't know how that that helps a lot.

[00:01:13]But in and also considerably stronger system than the piano arithmetic, the statement I made is correct.

[00:01:21]How does it compare to Graham's number?

[00:01:25]Well, if I remember Graham's number, I think it's kind of not noticeable. Yes.

[00:01:31]It's minuscule in comparison. It's an epsilon. Now you can make artificial You can play this artificial game in many ways.

[00:01:38]Uh but this is a particularly elegant one that is mathematically very uh friendly.

[00:01:47]If you remember the definition. Yeah, what is the definition of that number?

[00:01:51]Yeah, uh Okay, you look at you look at finite trees Okay, first there's an infinite fact that's behind it.

[00:02:02]And then you think of this as a finite approximate a finite form so to speak.

[00:02:07]And the infinite approximation is you have an infinite sequence of it goes back to JB Kruskal.

[00:02:14]Uh you have an infinite sequence of finite trees with uh three colors if you want. I do it with three colors. Uh with three colors on the vertices.

[00:02:26]Uh each vertex has one of the three colors. They have an infinite sequence of of finite trees, then one of the trees is a part of a later tree.

[00:02:37]And part of means there's a color preserving embedding that's in preserving. It's a it's a slightly technical notion, but it's a but it's the obvious notion of homeomorphic embedding. There's a there's a whole combinatorics that was well known of of of finite trees and how they fit together.

[00:02:55]What does it mean but you know, isomorphism and all that. Okay, so once again, and then given any infinite sequence of three colored trees, finite trees, sorry, finite.

[00:03:04]Uh one of the trees in the list is a is embedded in one of the later ones in the list. This is a famous theorem of JB Kruskal. And I noticed or maybe I wasn't the first to notice, but I did so I made the first to do something with it. Uh I noticed that this proof involved looking at all possible infinite sequences of trees.

[00:03:30]Uncommonly many. And and this proof was so far from being local.

[00:03:35]You might expect you just locally figure out if this piece is so it was so unusual this JB Kruskal proof. Yes.

[00:03:43]And I proved uh why it was so crazy. It was so strange. I proved that that Kruskal's theorem can't be proved in systems you'd expect it to be proved. Now, this is still nothing like ZFC. Is ZFC incompleteness is something entirely different a different level.

[00:04:01]But but but this is a micro incompleteness so to speak.

[00:04:06]And uh I So I kind of proved that you need uncountable sets to prove this Kruskal thing.

[00:04:14]Okay?

[00:04:16]Now, how do you approximate the Kruskal thing? Cuz after all, it's about infinite sequences.

[00:04:23]Well, it turns out that from the things like the compactness theorem and well known in mathematics Yeah. you can show that if you put a bound, if you say that the ith tree has at most i vertices, the ith tree has at most i vertices, so you put a bound on the grow on the growth rate of this trees, then you can find one of the trees is embedded in a later one.

[00:04:50]But you can even find a stopping place.

[00:04:52]You can even put a bound on how far you have to go up in order to know that. You follow me? In other words, you don't have to go a long way.

[00:05:00]So how far do you have to go up to get this?

[00:05:04]That's called tree of three.

[00:05:07]Tree of three says how long do you have to go on like this so that one of them is embeddable in a later one.

[00:05:12]As long as they're the ith tree as long as the ith tree has at most i vertices and we have three colors.

[00:05:19]That's what tree of three is.

[00:05:21]How how long do you have to go up there?

[00:05:23]Now for general mathematics, you can prove from Kruskal's theorem, which is about the infinite sequences, you can prove that there is a bounding place. There is a place so high that you can get there.

[00:05:38]So if you know Kruskal's theorem, you actually know tree of three.

[00:05:42]Hmm. Hi, everyone. Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my Substack. Subscribers get first access to new episodes, new posts as well, behind-the-scenes insights, and the chance to be a part of a thriving community of like-minded pilgrims. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So, click the link on screen here, hit subscribe, and let's keep pushing the boundaries of knowledge together. and enjoy the show. Just so you know, if you're listening, it's c u r t j a i m u n g a l {dot} org. kurtjaimungal.org.

[00:06:25]Okay, now what is this have to do with the relationship between infinity and the finite?

[00:06:31]Well, this says that there's really a very close connection between the really big finite, the outrageous finite, and the small infinite.

[00:06:41]The smallest infinity is just omega, the you know, the the the the size of the natural numbers.

[00:06:48]That's the smallest infinity. You know, Cantor has higher infinities, like the real number line and all that.

[00:06:53]>> Yes, yes, I have a video about that.

[00:06:56]Yeah, so so the smallest infinity is very closely associated and approximated by the outrageously finite, outrageously large finite.

[00:07:08]And we see this pattern over and over again.

[00:07:13]Now, I actually took this even more crazily.

[00:07:17]Put on my crazy hat, okay? I said, "Suppose I'm a finitist. I I say, 'Okay, only finite things, maybe big things, maybe not maybe not even I don't even like big things.' You know, if you're an applied computer scientist, you don't care about gigantic numbers. It's too impractical, right?

[00:07:36]Too impractical. You only care about things like a quadrillion or something is about as all about the most you could stand for. So, uh So, what if I put my finitist hat on?

[00:07:49]And I say, "Okay, all of mathematics I'm going to make finite."

[00:07:53]I know it's not finite directly, but I'm going to find finite approximations and thesis all of mathematical ideas can be are already represented in the finite.

[00:08:09]Pretty obvious idea.

[00:08:11]In other words, all this real number stuff, all these partial differential equations, even all this set theory stuff, large cardinals it's all fundamentally finite. But, uh this is my crazy hat. Yes. A finitist hat.

[00:08:27]>> [laughter] >> And in fact, uh the book embedded maximality has a section on purely finite forms which shows, in a sense, that all these large cardinals beyond ZFC uh can be thought of in finite terms.

[00:08:47]Now, let me ask you this. Yeah. What about ultrafinite terms?

[00:08:53]Okay, you talking about ultrafinitism?

[00:08:57]Well, that No, that's the point That's the That's the idea that the that you shouldn't go past uh very small numbers.

[00:09:05]That already 2 to the 100 is absurd.

[00:09:07]>> Mhm. That's ultrafinitism. Yes. That's part of the crazy hat.

[00:09:11]>> Interesting. Uh if you really want to push this really hard, all mathematical adventures can be properly imitated or realized in a computer screen. That level of of of detail. In other words, pixels with colors.

[00:09:31]Everything there is is is just pictures. That's already more than enough. Now, we kind of know that this is true in some sense cuz the human mind is thinking about things. I'm looking at at at the screen and I'm looking at everything. Everything I think about my my my my my work, my page, this is all well within the computer screen's size of information, which is only X bits, where X is you know, you can thousand bit >> Sure. whatever, right? So, this is actually the most radical.

[00:10:05]And now the question is all these fancy mathematical ideas we have, including large cardinals in ZFC, how do we actually say that we've actually uh properly represented them in a picture? I believe this can be done.

[00:10:22]I think that what's in my book is is nowhere near that.

[00:10:26]That powerful, but it does suggest some things.