TREE vs Graham's Number - Numberphile

[@numberphile]

Supporting #TeamTrees on a quest to plant 20 million trees - www.teamtrees.org/ Original brown papers from this video available to support the campaign - bit.ly/brownpapers

[@yubullyme2884]

Numberphile you should do tree 20 million

[@whatisthis2809]

Tree(20,000,000)

[@altfist]

Oh can you do a video on SCG(13)?

[@whatisthis2809]

*_ＷＨＹ ＩＳ ＴＨＥＲＥ ＩＮＦＩＮＩＴＥ ＦＩＮＩＴＥ ＮＵＭＢＥＲＳ?!_*

[@whatisthis2809]

More googology please :3

[@bigpopakap]

5:16 "You're giving the TREE more juice". This was the funniest, most succinct way to describe the same intuition I had!

[@dialecticalmonist3405]

I'm not sure what the term is for "rate expansion". For now "rate expansion" = juice.

[@flagmuffin1221]

Juicing the equation

[@adarshmohapatra5058]

I thought of it like this. The tree function gives much larger values than the g function. So at the enormous scales that we're talking about, all that matters is what is inside the tree function. So tree(g64) is better than g(tree(64)) because you're giving a bigger number to tree. It doesn't even matter what g is doing at this point.

[@bigpopakap]

...in other words, giving the juice to TREE, not g 😉. Give that tree more juice!

[@namelastname4077]

some would say he gave it more sauce, not juice

[@salientsoul]

19:15 - “if omega’s so great, why isn’t there an omega 2, huh?” 19:20 - “oh ok I’ll shut up now”

[@sugarfrosted2005]

Incidentally, this doesn't work for uncountable ordinals, like omega_2.

[@godoverlordquacken4003]

*Omega timea 2 wants to know your location*

[@want-diversecontent3887]

sugarfrosted Yeah, it only works up to ε_0. (ω^ω^ω^ω^...)

[@DWithDiagonalStroke]

Wut about cantor's ordinal?

[@PanduPoluan]

Omega acting all gangsta until Epsilon arrives.

[@sudoku0095]

A couple years ago, I planted a tree After one year, it was 1m tall After two years, it was 3m tall How tall will it grow in year 3?

[@scoutgaming737]

We are gonna die

[@petergriffinhentai4724]

Sit on top if you want to evade tax forever

[@it_genfailure]

* tree pierces the outer shell of the universe *

[@OllyGucci]

@@petergriffinhentai4724 lol

[@super-awesome-funplanet3704]

Can you please show me a spread sheet with the heights that the tree has had Not just exactly 1 year after you planted it and exactly 2 years after you planted it but also provide values between 0 years after you planted it and 1 year after you planted it and values between 1 year after you planted it and 2 years after you planted and all the way to now. Ps it should not be too hard to figure out the heights for whole numbers like 0,1,2,3,4... even if some of them you have to use a weird mathematical function to show the answer (Like even weirder than power towers.).

[@krozjr5009]

Remember this meme? Marvel: Infinity War is the most ambitious crossover in history. Numberphile: TREE(Graham’s Number).

[@MuzikBike]

Nah, let's do TREE(TREE(TREE(...TREE(g64)...))), where TREE is repeated G64 times.

[@sinom]

@@MuzikBike why stop there? why not repeat it TREE(G64) times? Or TREE(TREE(G64)) times?

[@Theboss24611]

Or just the crossover of Numberphile and Mr Beast.

[@simohayha6031]

@@sinom how bout ∞?

[@ganaraminukshuk0]

What if we planted TREE(g64) trees?

[@Alex_Deam]

"TREE vs Graham's Number" is basically clickbait for mathematicians

[@Adraria8]

I mean yeah it’s clickbate but in fairness they weren’t lying

[@Danilego]

TREE won by a landslide... A landslide of orders of infinities!

[@undercoverdetective463]

no coz if u know this its obvious whats bigger and u gain nothing new from the vid. but people who didnt knew can gain something

[@bsinita_wokeone]

I'm not mathematically smart.......but i due enjoy learning about big numbers and I mean BIG numbers like the ones larger than the ones in this video. Like Fish number, etc

[@edghe119]

The Gogeta vs Broly of the math world

[@GeoffBeggs]

So sure, Tree(Graham’s Number) is big. But I have just been exploring the harmonic series (1 + 1/2 + 1/3 + 1/4 + 1/5 …). It is divergent. It takes around 10^43 terms just to get it to sum to a hundred, and it gets way way slower after that. So my number (‘Geoff’s Number’ if no one has claimed this before) is: “The number of terms required for the harmonic series to sum to Tree (Graham’s Number)”.

[@prod_EYES]

Pin this comment

[@GeoffBeggs]

@Joji Joestar I’ll have to take your word on that. Sounds big.

[@rafiihsanalfathin9479]

@@GeoffBeggswell 1+1/2+1/3+...+1/n approximately ln(n)+euler m constant and the approximation gets better and better the larger the n. Because tree(g(64)) is so massive, e^(tree(g(64)-euler m constant) number of term is super sccurate approximation

[@StoicTheGeek]

@@GeoffBeggs yes, it is the same size as TREE(Graham’s number). When you are dealing with numbers this big, raising them to a power doesn’t make much difference.

[@craigdavies2598]

My number is:D(D3)3

[@MagruderSpoots]

If each of my brain cells was a brain, lets just call that an omega brain, I still wouldn't understand this.

[@BaldAndroid]

This makes my brain feel like it is a brain cell.

[@jonadabtheunsightly]

Yeah, but what if each of your brain cells contained as many brains, as your brain has brain cells? No, wait, what if each of your brain cells contained as many brains, as the number of possible permutations on the set of all brain cells in all of the brains in all universes real and imaginary? No, wait, what if all brains were like that, and then what if each of your brain cells could produce that many new brains per nanosecond, for each possible permutation on the set of all of the brain cells in all of those brains?

[@dahemac]

😂

[@U014B]

If I had a TREE(g(Γ₀!))-brain for every 1/(TREE(g(Γ₀!))) Planck Volume within the known universe (let's just call that a Ж-brain), and then had Ж Ж-brains for every one of those, I would probably die.

[@timothymonk1356]

@@jonadabtheunsightly Even if each of those brains had the combined capacity of the greatest scientists in the history of humanity, you still wouldn't come close to comprehending these numbers

[@CylonDorado]

Last time on Number Ball Z! Graham’s Number: “It’s no use, he’s too strong!” TREE (3) : “We have one option. We have to combine!”

[@nowonmetube]

@Nix Growham

[@omri9325]

It's not even his final form!!

[@omri9325]

It's over 9,000!

[@Redhotsmasher]

@@omri9325 WHAT 9000?!

[@imveryangryitsnotbutter]

@@omri9325 I mean, you are technically correct.

[@sean..L]

"But I don't need to stop!" He's gone mad with power.

[@anhbui-bc4ew]

don;'t

[@oz_jones]

*math

[@hardnrg8000]

@Nicholas Natale yes.

[@finnnaginnn]

I've gone madder.

[@PC_Simo]

Yes. Next, he’ll go mad with tetration. 😅😮😨😱🤯

[@3dtesseract853]

Every other KZbinr: "let's plant 20,000,000 trees!" Numberphile: “let's plant TREE(Graham’s Number)!”

[@AlabasterJazz]

Not enough matter in the conceivable universe to plant that many trees

[@ABaumstumpf]

i would highly advise against turning the entire observable universe into to strange matter with more than tree(3) trees in every possible location..... Also it would cost a lot of money.

[@Ken-no5ip]

BACHOMP There probably isnt enough quarks to reach that number

[@ABaumstumpf]

@@Ken-no5ip in the entire observable universe, filled to the limits of the pauli exclusion principle, would not be nearly large enough. Those numbers are just too insanely large.

[@theheckl]

that factorial at the end

[@Lucasinbrawl]

"Anything beyond gamma zero gets really messy." Yes, all was beautifully in order before then ;)

[@TheAlps36]

Ironic that they're called "ordinals"

[@chaohongyang]

I can confirm this, many post gamma zero notations are off the scale complex for new people to understand

[@The360MlgNoscoper]

Gamma gamma zero (;

[@j.hawkins8779]

@@chaohongyang actually, its ridiculously easy to go past it.

[@scathiebaby]

@@j.hawkins8779 Add 1

[@Darkness2179]

Man I love this guy's charisma, he's so genuine.

[@notmarr2000]

His book is amazing as well: "Fantastic numbers and where to find them."

[@fernandourquiza4593]

@@notmarr2000 can you like this comment just to remember myself to buy it?

[@notmarr2000]

@@fernandourquiza4593 the book is utterly mind blowing. I am half through (last chapter "Graham's Number, current chapter TREE (3)). The book is more than about math - he gets into a lot of physics, the concept of how big would the universe have to be before you would find an exact double of yourself, is the universe that big? Ect.

[@SyenPie]

@@fernandourquiza4593 4th like after 8 months just checking in if you bought it 😄

[@EGarrett01]

Now this video lives up to the name Numberphile.

[@MrBlaDiBla68]

Indeed, in math, chess, soccer and boxing, *drive* is important to "win" ;-)

[@TheFilipFonky]

@@MrBlaDiBla68 wot

[@Sakkura1]

Aleph-null bottles of beer on the wall, aleph-null bottles of beer, take one down, pass it around, aleph-null bottles of beer on the wall.

[@ThorHC11]

Best part is that "aleph-null" has the same number of syllables as "ninety-nine." So the rhythm keeps up!

[@naresu]

that's a lovely one

[@InsertPi]

unfortunately subtraction isn't defined for infinite cardinals

[@nate_storm]

Infinity (aleph null) minus one is infinity

[@pst9056]

Klein bottles?

[@iau]

It's crazy that such a simple "game" to explain, like TREE(n), which you may easily explain to even a first grader, is so insanely more powerful than even Γ₀, which requires pretty advanced mathematics to even begin conceptualizing. Mathematics is beautiful!

[@R3cce]

TREE(n) lies between the SVO and LVO in fast growing hierarchy. The SVO is lower bound and LVO the upper bound. It is much closer to the SVO but slightly faster than that

[@R3cce]

The SVO and LVO is just ridiculous just to let you know. If you want i can link a video to explain these ordinals. Then you will understand why Tony said in the video that anything beyond gamma gets messy😂😂

[@bdjfw2681]

@@R3cce sound fun , link pls.

[@AymanTravelTransport]

@@R3cce If you look at it through the levels of googology: level 0 is all the single digit and small numbers tending to zero; then level 1 goes from double-digits to a million; then level 2 goes beyond this all the way to a million digits (yes, googol and virtually every number we can practically deal with in our observable universe doesn't even get past level 2); level 3 starts entering the realms of tetration with googolplex and the likes; level 4 goes well beyond googolplexian and so on. You can see how going up just one level in the realms of googology gets to even greater magnitudes much further away than the levels before it. Well, G1 sits around level 6 or 7 and then G64 is much further at level 12, with fw(n) starting here or the level before it (11) let that sink in. So you can see that the difference between 12 levels is the difference between mere counting and shooting past G64 by iterating the number of arrows. Now you run through all the ordinals from omega through epsilon, zeta, eta until you reach the insane gamma-zero; the latter is all the way up to level 100 in the fast-growing hierarchy, meaning that counting to even G(G(...G(64))) [G64 iterations] would be much faster than using the G(n) function (or even f-epsilon0 for that matter) to reach the monsters produced by fgamma-zero(n). However, even then TREE(3) would laugh at this, as it's all the way up in level 120. That's right, an entire 20 levels ahead of fgamma-zero(n), which is insanely further apart than 0-1 and G64 which is merely 12 levels. This means even using fgamma-zero(n) to reach TREE(3) would be much slower (which is still an understatement) than merely counting to G64, even if you were to count in base 1/G64.

[@R3cce]

@@AymanTravelTransport According to Googology, the TREE sequence has the ordinal of (SVO times Omega) in the fast growing hierarchy

[@MilesEques]

"This is starting to terrify me now." "But I don't need to stop!"

[@jolez_4869]

ITS TIME TO STOP

[@ValexNihilist]

@@jolez_4869 laughed too hard at that

[@renanmaas3502]

That guy: Reaches an Unthinkably fast growing function that starts to bend the fabric of space-time. Also That guy: i CaN CArRy oN...

[@TheAlps36]

Please...please stop. In the name of sanity please stop

[@Parasmunt]

Don't go into the TREES stop stop.

[@GermaphobeMusic]

_looking at all the youtubers making tree videos_ "Oh yeah. It's all coming together."

[@jinjunliu2401]

although some trees were probably harmed due to the amount of brown paper used here

[@Snort70]

Hey it’s me you stole my comment cool idc

[@Snort70]

Germaphobe I don’t care tho

[@carbrickscity]

Nothing beats this one since pretty sure none of the others could come up with something like TREE(3)

[@johnathanmonsen6567]

This is absolutely the best explanation I've seen of just how much more massive TREE(3) is than g64.

[@AymanTravelTransport]

If you look at it through the levels of googology: level 0 is all the single digit and small numbers tending to zero; then level 1 goes from double-digits to a million; then level 2 goes beyond this all the way to a million digits (yes, googol and virtually every number we can practically deal with in our observable universe doesn't even get past level 2); level 3 starts entering the realms of tetration with googolplex and the likes; level 4 goes well beyond googolplexian and so on. You can see how going up just one level in the realms of googology gets to even greater magnitudes much further away than the levels before it. Well, G1 sits around level 6 or 7 and then G64 is much further at level 12, with fw(n) starting here or the level before it (11) let that sink in. So you can see that the difference between 12 levels is the difference between mere counting and shooting past G64 by iterating the number of arrows. Now you run through all the ordinals from omega through epsilon, zeta, eta until you reach the insane gamma-zero; the latter is all the way up to level 100 in the fast-growing hierarchy, meaning that counting to even G(G(...G(64))) [G64 iterations] would be much faster than using the G(n) function (or even f-epsilon0 for that matter) to reach the monsters produced by fgamma-zero(n). However, even then TREE(3) would laugh at this, as it's all the way up in level 120. That's right, an entire 20 levels ahead of fgamma-zero(n), which is insanely further apart than 0-1 and G64 which is merely 12 levels. This means even using fgamma-zero(n) to reach TREE(3) would be much slower (which is still an understatement) than merely counting to G64, even if you were to count in base 1/G64.

[@lookoutforchris]

I’ve got a large number I’m working on called ‘yo mama’

[@PTNLemay]

Brady's "more juice power" proof. I like it.

[@DFPercush]

Graham-ade, it's got what TREE craves!

[@bigpopakap]

it's rigorous enough for me!

[@PC_Simo]

So do I 🧃. P.S. You’re welcome for your 512th like. 👍🏻

[@PC_Simo]

@@DFPercush Exactly 👌🏻🎯😅.

[@PC_Simo]

@@bigpopakap Same here 😌.

[@kingbranden1369]

They pulled out ordinal collapsing functions on us. They really brought the big guns for this fundraiser.

[@AndrewBlechinger]

And yet they didn't get to Aleph-one

[@DarkestValar]

A Large countable ordinal, but not quite an ordinal collapsing function.

[@imperialguardsman135]

Ordinal what?

[@Peter_Schluss-Mit-Lustig]

Well they didn't even talk about fundamental sequences

[@ig2d]

It's all about the juice

[@hylen26]

"This next guy, I'm not going to write it out, because it has 121 million digits." This has to be in the top ten Numberphile videos of all time. Maybe top three even?

[@nocktv6559]

Top TREE

[@joshuamiller5599]

“Well, the problem is that you’re just dealing with finites.” This problem is indeed found in so many situations.

[@etfo714]

Newton/Leibniz be like this when inventing calculus.

[@antonhelsgaun]

A problem when looking at my account balance

[@dAvrilthebear]

I encounter this problrem when paying for my gaughter's tutors)

[@douche8980]

Sounds like a racist statement :(

[@MikeRosoftJH]

Aleph-0 green trees growing at the sprawl. Aleph-0 green trees growing at the sprawl. And if one green tree should accidentally fall, there are Aleph-0 green trees growing at the sprawl.

[@TheTwick]

I remember, on the schoolyard, when the biggest number was “a BAZILLION”🤯

[@xexpo]

@@boudicawasnotreallyallthat1020 I don't mean to obliterate you.. but I raise you 2 bazillion.

[@teriww]

....2 bazillion plus infinity🙀🙀🙀🙀

[@user-fk6cb9en8v]

@@xexpo 2 bazillion-fantastillion

[@wallonice]

I remember it being "uncountable"

[@BlokenArrow]

Or a Brazilian

[@grugruu]

This is the most intense AND my favorite part of this whole channel.

[@emoglobin2195]

Is it me, or does 20 million suddenly sound like a pathetically small number

[@sadhlife]

time to plant TREE(3) trees

[@anixias]

Time to plant TREE(TREE(TREE(....tree(64) times...))) trees

[@yvesnyfelerph.d.8297]

120million digits sounds like nothing at all, given what they are looking at

[@DirtyRobot]

That's basically a day's worth of disposable chopsticks in China. Thanks internet, Now Chinese can enjoy eating for an extra day.

[@schenkov]

Actually first thing I thought when I heard about that project was:"20 million threes are not so much at all"

[@BedrockBlocker]

The TREE function impresses me everytime. It's so simple yet it blows everything away.

[@knightoflambda]

Just wait, one day they'll finally explain the Busy Beaver function BB(n), which grows so fast there literally cannot exist a function that can compute any of its digits. It's insane just how fast it grows. I heard that even getting a lower bound on BB(20000) is impossible in ZFC. Of course, BB(n) is tiny compared to its relativized cousins. And we aren't even out of the lower attic yet. In the middle and upper attic, there are numbers so large that you need to add extra axioms to ZFC in order for them to exist.

[@yogaardianto2269]

@@knightoflambda what is the most faster growing fiction in googology?

[@pierrecurie]

@@knightoflambda I think they already did an episode on BB. Scott Aaronson proved that computing BB(~8000) requires proving the (in)consistency of ZFC (basically brute forces some statement that is true IFF ZFC is consistent).

[@purpleapple4052]

@@knightoflambda they mentioned and explained some Busy Beaver stuff in the video about Rayo's number

[@isuller]

@@knightoflambda actually it is true that BB(n)>TREE(n) for n>k for some k value. But my guess is that "k" is huge itself - I mean it may be bigger than Graham's number. So while it is true that BB is a faster growing function than TREE it doesn't mean that in the region of "normal" numbers BB(n) is bigger than TREE(n) :-)

[@felooosailing957]

Fascinating that g and TREE are so fast growing that you need transfinite ordinals to put them in a hierarchy. This is probably the best way to convey their power.

[@thurston2235]

The paper change is the real reason we watch this channel.

[@BobStein]

Yep. That joke's got layers, man.

[@AndrewTyberg]

Ummm... Not true....

[@pleasuretokill]

It's the one thing here I can comprehend

[@Ishub]

@@pleasuretokill same

[@waldothewalrus294]

The jingle on it keeps me living

[@jonipaliares5475]

Never thought transfinite ordinals could be useful with something finite like sequences of integers. Amazing video!

[@martinshoosterman]

Oh man. You should look up the proof of Goodsteins theorem, using trans finite ordinals. Its a statement about sequences of numbers which is proven using ordinals.

[@watcher8582]

All the ordinals that were mentioned in this video were still countable, i.e. they can be viewed as representing a (non-standard) ordering of the natural numbers. That is to say, the transfinite ordinals play the role of intrudcing jumps (in this case the jump is taking the diagonal in the constuction of f's). As such, the cardinalities of any of those ordinals is N, and thus all still smaller than that of the reals R.

[@Lexivor]

@@martinshoosterman Goodstein's theorm is fun. The function that calculates the length of Goodstein sequences has an ordinal of epsilon_0, much bigger than Graham's, but nothing compared to TREE.

[@grantchapman640]

21:17 you can’t fool me, you’re just drawing squiggles now

[@TheDanksNewGroove]

Even when ignoring the awesome fundraiser, I think this is the coolest video you guys have ever made. Talking about stupidly giant numbers with no physical significance just because it’s fun. I love it, congratulations.

[@dvkprod]

Recommended reading for the course - Vsauce's How to count past infinity.

[@NoriMori1992]

Dyani K. Seriously. That video's the only reason I had the slightest understanding of the omega stuff.

[@zmaj12321]

If I haven't already seen that video I would have no clue what I was watching.

[@gdash6925]

Yea that inspired me to watch this numberphile video.

[@billvolk4236]

Vsauce, where we give disingenuous answers to clickbaity loaded questions without ever explaining what's fundamentally wrong with them.

[@dvkprod]

@@billvolk4236 dude, what is your problem

[@Uranyus36]

It's amazing that even without the ordinal Mathematics, we can still tell that TREE function grows (way) more quickly than Graham's function. TREE(n) literally goes from 1 to 3 to something that is way way way way way bigger than Graham's number, while G(n) needs 64 layers to go from 3^^^^3 to Graham's number. It's absolutely safe to say that at least the numbers G(1) to G(64) are all within the gap between 3 and TREE(3). The jumping between G(n) is essentially stationary compared to that between TREE(n).

[@PC_Simo]

Exactly 👌🏻.

[@caringheart34]

G(0) is also 4 so basically the entire graham sequence

[@PC_Simo]

@@caringheart34 I thought the same thing 🎯.

[@R3cce]

@@PC_SimoTREE(n) grows at a rate between the SVO and LVO in fast growing hierarchy. These ordinals are beyond gamma. I can link a video to explain these ordinals if you want. You will then understand why Tony said in this video that anything beyond gamma gets messy😂

[@Empiro3]

Things can start slowly then get really big later though. Tree is still a computable function. The Busy Beaver function has pretty reasonable values for small values, but it grows much faster than any computable function.

[@juliankneaz6893]

The mathematicians went out of control, somebody please stop them

[@geekjokes8458]

NEVER

[@EpicMathTime]

no

[@FrankHarwald]

Not their fault - one of them SUPER busy beavers outta control!...

[@otakuxgirl6]

No

[@TheAngelsHaveThePhoneBox]

12:28 My brain just collapsed into a black hole. Edit: Now after seeing the whole video, my brain collapsed into so many black holes that the number of black holes itself collapsed into a black hole and then another black hole and this happened so many times that the number describing it also collapsed into a black hole.

[@goutamboppana961]

and so onnnn

[@adwitraj4923]

P O T A T O

[@kylebroussard5952]

I love how mathematicians get to a point where they're so smart they start making up numbers a 5 year old would spout off and then act profoundly amazed by a finite number within infinity.

[@homer4340]

Mathematicians after creating the number galleohalivitoxipityisnlotopiscisis22: 😮

[@jetzeschaafsma1211]

David Metzler has an excellent 40 part series on the fast growing hierarchy, ordinals and much much further.

[@michellejirak9945]

I thought this was a joke until I looked it up. Well, now I know what I'll be doing for the next month.

[@OrbitalNebula]

There's also Giroux Studios

[@chaohongyang]

@@OrbitalNebula And you, btw you need to make more FGH vids, they are so damn gud

[@OrbitalNebula]

Oh yeah. I'm now actually on the progress of making the next big numbers vid. It's just taking me quite long to make.

[@chaohongyang]

@@OrbitalNebula i fully support you, do whatever you want at your own pace homie :)

[@hewhomustnotbenamed5912]

This is literally the biggest collaboration in KZbin history. And it's for the best possible cause. I'm genuinely proud of this community.

[@erik-ic3tp]

Me too. This's a 10 out of 10 for Humanity today.

[@googleuser7771]

@@erik-ic3tp is 20 million trees a lot of trees?

[@erik-ic3tp]

Google User, Yes.🙂

[@Sam-ep7sc]

I love Numberphile. Hugely over my head but inspiring to know that there are people in the world who can come up with these concepts, and those who can put them across to the lay person.

[@tspander]

So nice to see so many channels contribute to #TeamTrees

[@RedDesertRoz]

I'm at just over 14 minutes and am going to have to rest my mind and finish this tomorrow. Have just watched the 2 videos on tree(3) beforehand. This feels like staring into the abyss and it's rather terrifying, and as well, my mind feels like it's melting down from struggling to comprehend such enormity. Who knew that maths could get kind of terrifying?!

[@ShahromUK]

The fact that Tree(G64) is still smaller than Rayo's number is just crazy

[@jimsteinmanfan80]

Why is that? He covered how Tree(G64) is defined in just a couple of videos. Rayo's number of course can describe something much more complex.

[@armityle29]

This was geniunely one of my favorite videos ever to have been uploaded to this channel.

[@PC_Simo]

I fully agree 👍🏻.

[@limbridk]

For sure one of the best videos on my favorite channel. Such elegant insanity. Love it!

[@bryanc1975]

I read a cool description of Graham's number somewhere, in terms of trying to picture it in universal physical terms. If my memory serves me, it went like this: It said that even the integer describing the number of digits in Grahams number could not be represented if you made every particle in the universe a digit, and the same would be true for the number of digits in THAT number, and even if you went down that "number-of-digits-in-the-previous number" scale, with each level down being represented by a single particle in the universe, you still would not able able to fit it into the known universe. I wish I could find that again.

[@r.a.6459]

In fact, g(1) itself, defined as 3↑↑↑↑3, is bigger than googolplexplex...plexplex (with googolplex 'plex'es)

[@vokuheila]

In fact, the number of digits in Graham's number is approximately Graham's number...

[@hurricane3518]

its on wikipedia

[@Veptis]

in the first 3 hours they are past 1 Million, hope this keeps afloat for a while

[@erik-ic3tp]

It's mind-blowing what crowdfunding could do if done right.

[@Veptis]

@@erik-ic3tp it's a giant collaboration, so that is unprecedented.

[@pluto8404]

@@Veptis "collobaration" you mean the 1% sit back and take all the credit while their followers donate all the money.

[@Veptis]

@@pluto8404 no, if it weren't for those people to initiate it and produce unified content on the topic. such an effort wouldn't be possible uncoordinated.

[@pluto8404]

@@Veptis I suppose we do need a large unification to combat all the carbon their Manson's and sports cars put out.

[@jwdeepsky]

So I've heard before that tree(3) was unfathomably bigger than graham's number, but damn. This is so far beyond an understatement.

[@SoleaGalilei]

I'm no mathematician, but thanks to your past videos I laughed out loud when I saw what this one was about, knowing we were in for another round of "STUPID big"!

[@zmaj12321]

Best Numberphile video in a while, but NOT for the faint of heart.

[@KYZ__1]

These big number videos make me unimaginably excited...

[@spyguy318]

I remember the VSauce video on Ordinal Numbers and Infinities; I was prepared for this one. Still amazing that TREE grows even faster than that!

[@lordheaviside2605]

Your original videos on Graham’s number are what got me so into googology in the first place. I can’t express how incredible it feels to see a Numberphile video on the fast-growing hierarchy! I love your videos so much!

[@denverbax6329]

22:51 Yoooo that is actually scary. I knew TREE was big, but I did not expect that.

[@R3cce]

TREE(n) is believed to grow at least as fast as the Small Veblen Ordinal or SVO for short. SVO is beyond Gamma in strength

[@jamx02]

@R3cce It more than likely isn’t anywhere close. SVO just covers a lot of area within ordinal collapsing functions so it more than likely grows faster than TREE(n), it’s just nobody really knows so they slap it on SVO because it’s the best estimate. The only thing we do know is it is between the Ackermann ordinal (Fefermann-Schutte fixed point) and the small Veblen ordinal.

[@coreyburton8]

You have combined my two favorite numberphile videos! Thank you!

[@00blaat00]

I love the hint of fear that trickles through his enthusiasm when discussing the functions over Gamma-Naught: "We must tread lightly here, lest we disturb the Old Ones who dwell in these regions..."

[@Zwijger]

It was quite intuitively obvious to me that Tree(n) was way bigger than g(n), the best way I can describe is that 3 is the first number in the Tree sequence to unlock it's full power, as you always have a first sacrificial colour, so you're kinda playing the game with n-1 colours. 0 colours for n=1 obviously stops, 1 colour for n=2 also has to fundamentally stop really quickly, but for n=3 you finally have 2 colours to play with. If 2 colours already gives the illusion that Tree(3) might be infinite at first glance, and remember this is the first "real" amount of colours to unlock the Tree game, then it only follows that this graph is exploding quicker with any more colours to play with from that point than anything you can make with normal iterations of mathematical functions, no matter how awesome a way you have to write them to become really big.

[@PC_Simo]

Also, you only have to climb up to the 3rd branch of the TREE-function to already be off-the-scale massively higher, than g(64), which is the 64th rung on Graham’s ladder.

[@AymanTravelTransport]

@@PC_Simo If you look at it through the levels of googology: level 0 is all the single digit and small numbers tending to zero; then level 1 goes from double-digits to a million; then level 2 goes beyond this all the way to a million digits (yes, googol and virtually every number we can practically deal with in our observable universe doesn't even get past level 2); level 3 starts entering the realms of tetration with googolplex and the likes; level 4 goes well beyond googolplexian and so on. You can see how going up just one level in the realms of googology gets to even greater magnitudes much further away than the levels before it. Well, G1 sits around level 6 or 7 and then G64 is much further at level 12, with fw(n) starting here or the level before it (11) let that sink in. So you can see that the difference between 12 levels is the difference between mere counting and shooting past G64 by iterating the number of arrows. Now you run through all the ordinals from omega through epsilon, zeta, eta until you reach the insane gamma-zero; the latter is all the way up to level 100 in the fast-growing hierarchy, meaning that counting to even G(G(...G(64))) [G64 iterations] would be much faster than using the G(n) function (or even f-epsilon0 for that matter) to reach the monsters produced by fgamma-zero(n). However, even then TREE(3) would laugh at this, as it's all the way up in level 120. That's right, an entire 20 levels ahead of fgamma-zero(n), which is insanely further apart than 0-1 and G64 which is merely 12 levels. This means even using fgamma-zero(n) to reach TREE(3) would be much slower (which is still an understatement) than merely counting to G64, even if you were to count in base 1/G64.

[@KhalidTemawi]

One of the best videos of Numberphile!

[@HeroDarkStorn]

KZbin: Let's all talk about trees. Numberphile: Challenge accepted

[@srizic1136]

Vsauce: What's the biggest number you can think of? Me a googol Numberphile: Tree(g(64))

[@Eric4372]

A googolplex factorial to the power of a googolplex squared, times Graham’s number!

[@srizic1136]

@@Eric4372 To the tree(g(64))th tetration

[@zzasdfwas]

But you can always define something bigger by iteration. Tree(tree(g(64))). Or tree(tree(...n times) (g(64))). And then you can iterate that.

[@haeilsey]

zzasdfwas and then you develop notation to recursively iterate the iteration, like idk Conway chain arrows on steroids in hyperspace as the gods

[@FlowiYT]

Me: TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(TREE(g64)))))))))))

[@Megamegalomane92]

You can go beyond gamma zero. f gamma zero: "This isn't even my final form!!!"

[@martinshoosterman]

Yeah. As far as I know you can go as far as f ω₁ ie, you can have f of anything smaller than ω₁ but you cannot define f for ω₁

[@donandremikhaelibarra6421]

@@martinshoosterman yes but you surely can’t have an f of an inaccessible cardinal right?

[@martinshoosterman]

@@donandremikhaelibarra6421 you can't even do f(omega_1) much less an inaccessible cardinal.

[@donandremikhaelibarra6421]

@@martinshoosterman is the inaccessible cardinal bigger than an infinite amount of alephs nested together?

[@NoahTopper]

The amount of times I just yelled "No way!" alone in my room is only slightly embarrassing.

[@austinlincoln3414]

lol

[@blipmachine]

"You're giving the tree less juice there but more juice here" The cameraman really gets the limitations of my brain power 😂

[@non-inertialobserver946]

TREE(g64): exists g(TREE64): Finally, a worthy opponent. Our battle will be legendary

[@jolez_4869]

TREE(TREE(3)) joins the game

[@AndrewTyberg]

But the second number is basically 0 compared to the first number.

[@mapari00]

This could be perfectly fine in the context of the quote, as tai lung thought he would defeat the dragon warrior, but in fact got stomped as if he was nothing. Later in the fight: “The Wu Shi finger hold?!?!?! Shi Fu didn’t teach you that!!!!!!!” “Nah, I figured it out. Scadoosh!!”

[@isaacwebb7918]

@@jolez_4869 Even TREE(TREE(3)) won't match SSCG(3). SSCG is for 'simple sub-cubic graph,' and it works similarly to the tree problem and resulting function, except there are fewer rules for simple sub-cubic graphs, making more graphs possible, and therefore (much, much, much...) longer sequences. SSCG(n) forms a similar sequence to TREE(n) (in that it describes maximum lengths of non-repeating sequences for a given number of tags, and in starting small and exploding by n=3), but outpaces it easily -- SSCG(3) is greater than TREE(TREE(TREE(TREE(TREE(...TREE(3)))))) -- if you nested that TREE(3) layers deep. TM,DR (Too math, didn't read) -- there's always a bigger function.

[@jolez_4869]

@@isaacwebb7918 Wow damn. Thats interesting!

[@innerufomaker]

Loved this video. The like button wasn’t enough for me. I’ve always used to do this sick thing of imagining very very big numbers, steps and distances since I was 5-6 y/o and it got to a point that I had to stop doing that. This video made me feel a part of my life which I’ve never been able or tried to share with someone else. I reply “speed” when I’m asked about my favorite thing in the world, and they think I just like to drive fast. In fact, I mean exponential growth of exponential growth of .... .

[@geekjokes8458]

I understand that feeling very well... im not sure about it being my favourite, but i do get excitedly anxious, it kinda hurts, about this sort of big...ness It's so unthinkably big, profoundly and absolutely indescribable... art, it seems, like the cosmic horror style of storytelling, is the only thing that can "properly" assign some meaning to this feeling, maybe precisely because it forgoes logic. Art, and mathematics.

[@erik-ic3tp]

iUFOm, Same for me too.😊

[@NoriMori1992]

That's beautiful.

[@NoriMori1992]

Have you ever watched the Vsauce video "How To Count Past Infinity"?

[@erik-ic3tp]

NoriMori, I’ve watched it yes.🙂

[@mAximUm123451]

18:50 "this terrifies me... but I don't need to stop!" A true classic

[@yuvalne]

People: there's no way Numberphile can join the #teamtrees thing Numberphile: hold my beer

[@masterimbecile]

Hold my Klein bottle.

[@yuvalne]

@@masterimbecile darn it should have seen that joke

[@TheScarrMann]

With the amount of paper used in these videos I'd be shocked if they didn't

[@triqky9301]

Well he did use paper...

[@trevorx7872]

Hold my juice

[@leo17921]

20:40 funny how its called epsilon 0 cause usually epsilon is used for small numbers

[@B0b0K1w1]

MILDLY INTERESTING

[@pandabearguy1]

Thats \varepsilon

[@happygimp0]

"512, quite big number" 7:10

[@LeBronJames-sj7ds]

LOLOLO

[@dylanmcadam8509]

Compared to the number in this video there is like no difference between 512 and -googleplex

[@fractlpaca6285]

This reminds of dreams I have when I have a fever... A tiny point would suddenly explode to gargantuan size, then compound upon its own size, until it filled my mind. Or marbles would arrange themselves into huge sparse, patterns while multiplying all the time....

[@36jbf-as031lb]

Newer thought there would be something so out of world and so "same" at the...same...time.

[@drumroll7073]

U are not alone. Also this dream has a very bad taste. i hate this dream for no reason...

[@jpdemer5]

That's why I stopped doing drugs.

[@xyz.ijk.]

I looked up how to express TREE(3) in terms of Gx. Here is the lower bound: G3[187196]3 (compared to G3(64)3). No wonder the growth is so astounding with TREE(x).

[@Dexuz]

That doesn't seem right at all, you can't express the value of TREE(3) in a function that grows infinitely slower than the TREE(3) function, just as you can't express the value of Graham's Number in any F(finite ordinal).

[@R3cce]

@@Dexuz TREE(n) is between the SVO and LVO in fast growing hierarchy

[@xyz.ijk.]

​​@Dexuz I agree, but I'm reporting what I looked up, I wouldn't dare claim to have calculated such a thing!

[@thefirstsurvivor]

theres no proof it's between those@@R3cce

[@gedstrom]

Tree(3) may be universes beyond G64 in size, but G64 is a LOT easier to understand how it is generated, even though we can't even begin comprehend its size. I can't even begin to comprehend how Tree(3) is computed!

[@callumsylvester9921]

Finally, a worthy opponent! Our battle will be legendary!

[@methyllithium323]

At this point, you can't even compare g(TREE(3)) even with TREE(4) because of how much faster TREE grows

[@jamx02]

TREE(4) is much larger than g(g(g(…g(TREE3)..) TREE(3) number of times

[@eightdogstreet]

@@jamx02WOW

[@jimsteinmanfan80]

Which is what fascinated me so when I learned of the Busy beaver functions that are so simple to describe yet grows so much faster than TREE.

[@neotaharrah6478]

This is one of the most mind blowing mathematical things I have ever seen. This is completely outrageous!

[@illogicmath]

Making all these videos Brady practically became a mathematician.

[@PickyMcCritical]

21:00 Right about here he starts foaming at the mouth. I love this guy lol

[@joostinatortje]

Well, it's allready a stated fact at the beginning of the video that TREE(3) is bigger than grahams number. This means that it grows beyond Grahams number in just a matter of 3 steps of n. (Of which the first step is not even relevant) This already shows that the growth of the TREE sequence is far bigger compared to Grahams sequence. Using grahams number as input for the TREE sequence would therefore always result in a higher integer than using the TREE(64) in Grahams sequence.

[@danielstephenson7558]

And as everyone knows we get Omega-3 from fish. So this video is telling me: plant Trees using fish.

[@DFPercush]

a herring!

[@danielstephenson7558]

@@DFPercush *jarring chord*

[@yaboi6851]

i hate this joke

[@parkerwest6658]

Just got flashbacks to the vsause vid about ordinal numbers

[@SpektralJo]

But his video was about cradinals

[@twigwick]

same lol

[@naresu]

Was reminded about aleph

[@lgbfjb7160]

Im terrible at math but I'm facinated at how incomprehensible these numbers are and how i still feel that somehow i could fathom it knowing i never will.

[@jj.wahlberg]

Ah the iconic “Paper Change” music returns

[@syedrahman5837]

This may be the most amazing Numberphile video. I have been completely humbled by it. Before watching this I used to think of numbers like G64 and Tree3 and thought incredibly large numbers can be created by recursions on Tree3. But the aha moment was to realize the generalization of the growth rates and that the rates are more interesting than any large number. The fact that G64 shows up do early in that sequence of series while Tree growth cant even be caught up by ordinal infinities and then some. Wow

[@R3cce]

TREE(n) grows faster than SVO but slower than LVO in fast growing hierarchy.

[@R3cce]

SVO stands for Small Veblen Ordinal and LVO Large Veblen Ordinal

[@modernwarriorsystems7347]

When I was in engineering classes, I would have LOVED to have him as my teacher.

[@wafflefox6393]

Oh boy oh boy! I already liked this video, but I'm gonna watch it later when I can give it full attention. Tree v Graham's number, yis!

[@naedolor]

Graham's number is so big that it can contain universes in itself. We might be living inside a Graham's number universe right now

[@dekippiesip]

It's overkill for the entire concept of universes. 10^100 is already enough to deal with universe scales compared to human or even atomic scales. You don't need that many powers of 10. Since the limits of the universe can be reached quickly by exponentiation we only need a sequence growing like f2 to quickly hit it's restraints. Tetration or f3 is enough to deal with the numbers associated with combinatorics questions applied on the universe, like in how many ways you can arrange all atoms in the universe and questions of that nature. Graham's number is simply waaayyy waayyyy beyond all of that. Even g(1) itself is complete overkill in that regard.

[@valthiriansunstrider2540]

G64 is way bigger than that, it could contain more observable universes than there are combinations and states of elementary particles in our observable universe, to such a vast degree that that description becomes nonsense in trying to express how big it is.

[@dekippiesip]

@@valthiriansunstrider2540 yeah, there is simply no 'real world' context you could ever use as a reference for that number.

[@Crazytesseract]

@@dekippiesip There could be universes with dimensions equal to Graham's number. There could be a Graham number of multiverses. How do we get to know?

[@knxcholx]

@@valthiriansunstrider2540 no way are you all thinking numbers contain the universe.........

[@BigDBrian]

Half this video is just Tony trying to find words to express the magnitude of these sizes, and we love it

[@natheniel]

4:24 I didn’t know I miss the paper change so much until I see one

[@jaimanparekh4616]

What’s the biggest number? Numberphile (starts rubbing hands together): My time has come

[@PXKMProductionsGaming]

I'd love to see more explanation videos on these higher level infinities. Also, despite being messy, I'm so curious about what stuff comes after Gamma Zero (or f(gamma zero)! I come back to this video a lot. how big numbers can get is so interesting to me.

[@R3cce]

The Small Veblen Ordinal (SVO) is the next ordinal after Gamma zero. After the SVO comes the Large Veblen Ordinal (LVO)

[@andrew_cunningham]

At this point my entire subscription feed has been replaced by trees. I guess I'm okay with that.

[@illogicmath]

This really exceeds my ability to comprehend.

[@DFPercush]

Don't worry, it exceeds the physical universe's ability to comprehend too.

[@Briekout]

How do folks wrap their brains around adstracts like this? So cool!

[@fuseteam]

"tree is off the charts" brady: wanna learn more about trees?

[@LukePalmer]

17:00 - 22:00 is literally just 5 minutes of woah massive HuUgGeEeE wowowowow gamma! alpha!! epsilon OF epsilon!!! UNIMAGINABLY you just can't even WOW it's MATH!!!!!!

[@CalvinHikes]

Isn't it still a small number though? I mean it's hard to imagine but it's a lot closer to 0 than it is to the infinite numbers larger than it. Relative to all numbers, it is a very small number. It's just a number that is larger than we have need for use of.

[@doicaretho6851]

@@CalvinHikes Yes, it's smaller than almost every other number

[@TheJulianmc]

@@doicaretho6851 Dont think so, since its defined beyond the cardinal numbers.

[@Xomage999]

So basically every time someone talks about power levels on DBZ.

[@tabeshh]

@@doicaretho6851 Does that mean every single number is relatively small?

[@jonciobanu4546]

Just the fact that no finite f(n) hierarchy could describe the growth rate of Graham's number, let alone TREE(n), blows mind mind. Truly shows how unimaginably large those numbers are.

[@sejdatalukder6798]

one thing that i find interesting is that tree(65) is already way bigger than g(tree(64))

[@AaronSmith1]

I'd really like Numberphile to do at least one more video on TREE(3). Specifically, I still don't quite get how it grows so quickly. Maybe if I saw more examples using 3 seeds I'd get it? Not sure. It seems like either it should be infinite or the number would be smaller with 3 seeds. Graham's Number seemed a lot more logical in the way it's built up. With TREE(3) I still feel like I'm being asked to just take it on blind faith that it's really, really big.

[@R3cce]

Even TREE(4) is bigger than GGG….G(TREE(3)) where the number of G iterations is TREE(3) itself. Just to show how fast TREE(n) grows

[@R3cce]

Ever heard of SSCG(3)? It’s even bigger than TREE(TREE(…..(TREE(3)) with TREE(3) iterations of TREE

[@aliroostaei9122]

These videos are so amusing to watch, even for a nerdy med student❤

[@kerbodynamicx472]

I’m kinda scared when a toddler says “I can count two trees!”

[@xerosfs]

_”I can count three trees!”_

[@xerosfs]

@Lakshya Gadhwal lol