The Enormous TREE(3) - Numberphile
Why Hacker News recommends it
HN commenters cite it as the go-to accessible explanation of TREE(3) and as part of Numberphile's series on enormous numbers, often when Graham's number or Busy Beaver values come up.
Recommended by 2 people across 2 different months, 2018 to 2026.
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when asking or answering questions about the TREE function, especially TREE(2), TREE(3) and TREE(4), and how large these values are.
Recommended by 0 people across 0 different months, 2018 to 2020.
What it is
Numberphile video in which Professor Tony Padilla explains TREE(3), a finite but astronomically huge number arising from a combinatorial game on trees related to Kruskal's tree theorem. It is presented as a conversational explainer with whiteboard-style drawings, and it dwarfs Graham's number.
What people said
Strong rec jiggawatts · 2024-05 · Hacker News · on “A Trial HIV Vaccine Triggered Elusive and Essential Antibodies in Humans”
If math can make you feel awe, you've got to watch the Numberphile series on very big numbers. They're constants used in proofs that are specific, well-defined integers known to be finite, but so huge that the physical universe does not have enough particles (or even Plank-scale microstates) in it to represent the number of digits these numbers have! Absolutely mind-bending stuff. Listed in increasing order: "Graham's Number" [LINK] "How Big is Graham's Number? (feat Ron Graham)" [LINK] "The Enormous TREE(3)" [this video] "TREE(3) (extra footage)" [LINK] "The Daddy of Big Numbers (Rayo's Number)" [LINK]
Recommends bionoid · 2018-01 · Hacker News · on “Graham's Number Is Too Big to Tell How Big It Is”
I will take this opportunity to promote the YouTube channels of Brady Haran[0], Numberphile specifically. It has all sorts of interesting videos, including Graham's number [1] (one of the videos featuring Ron Graham) and TREE(3) [2]. His other channels are worth checking out too, Sixty Symbols and Deep Sky Videos being my favorites. To be clear, I have no affiliation with this, just a fan. [0] [LINK] [1] [LINK] [2] [this video]
Cites Carl Mummert · 2018-02 · Mathematics Stack Exchange · on “What elementary theorems depend on the Axiom of Infinity?”
The axiom of infinity, in ZFC set theory, says that there is an infinite set. Once you have one infinite set, because you can take the powerset of any set, you can form larger and larger infinite sets, in terms of cardinality. On the other hand, none of the other axioms of set theory is able to create an infinite set out of finite sets. So the axiom of infinity is important as a fundamental basis in set theory, just as an axiom that there are infinitely many different points is important in Euclidean geometry. Otherwise, we could have a world in which all sets are finite, or a geometry in which there are only finitely many points. Those kinds of models are interesting but are not the main su
Cites Oddthinking · 2018-10 · Mathematics Stack Exchange · on “Why isn't TREE(2) = 6?”
I was just introduced to the TREE() function, by a 2017 Numberphile video [this video] . In it, Tony Padilla explains a simplified version of the calculation of the TREE function: We are building a forests with as many trees as we can, following certain rules. Each tree in the n th forest can't have more than n nodes, and each tree in a forest may not "contain" an earlier tree from the same forest (i.e. it can't be "inf-embeddable"). The maximum number of trees in a forest is TREE(n) . On the way to explaining that TREE(3) is very large, he computes the size of TREE(2) as 3, giving this sequence for the forest (where G is a green node, and R is a red node): G R-R R However, following the rules as
Cites Joey Peters · 2019-01 · Mathematics Stack Exchange · on “explicit upper bound of TREE(3)”
TREE(3) is known to be finite, but few finite upper bounds have been proven that do not rely on constructions such as TREE(n) and n>3. As for lower bounds, saying $g_{64}$ (Graham's number) is a lower bound for TREE(3) is like saying 1 is a lower bound for Graham's number. Even $g_{187,196}$ has proven to be an EXTREMELY WEAK lower bound for TREE(3). See [LINK] and [this video] .
Cites QWERTYL · 2020-06 · Mathematics Stack Exchange · on “Is $TREE(4)$ bigger than $TREE(3)$”
I've recently heard about the enormous number [LINK] $TREE(3)$ in a [this video] , and I was wondering if $TREE(4)$ would be bigger?
Cites deanCommie · 2026-10 · Hacker News · on “BB(10) > Graham's Number”
IMHO an output of a function being bigger than Graham's number isn't particularly impressive. Consider TREE(3) -> [this video] Graham's Number is notable for being one of the first, being an actual relevant bound in an unrelated proof (not just the output of a function), of teaching people the Up Arrow Notation. To me, BB(10), TREE(3) etc. being so large is only minorly more impressive than GG(1) where I define I define GG(x) as "Graham's Number + X"