TREE vs Graham's Number - Numberphile 

Numberphile you should do tree 20 million

Tree(20,000,000)

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

*_ＷＨＹ ＩＳ ＴＨＥＲＥ ＩＮＦＩＮＩＴＥ ＦＩＮＩＴＥ ＮＵＭＢＥＲＳ?!_*

More googology please :3

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

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

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

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

The Gogeta vs Broly of the math world

We are gonna die

Sit on top if you want to evade tax forever

* tree pierces the outer shell of the universe *

@@petergriffinhentai4724 lol

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.).

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

Juicing the equation

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.

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

some would say he gave it more sauce, not juice

don;'t

*math

@Nicholas Natale yes.

I've gone madder.

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

time to plant TREE(3) trees

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

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

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

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

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

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?

😂

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.

@@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

Pin this comment

The amount of terms needed is approximately e^(Tree(Grahamsnumber)-gamma) where gamma is the euler mascheroni constant

@@hybmnzz2658 I’ll have to take your word on that. Sounds big.

@@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

@@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.

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.

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

*Omega timea 2 wants to know your location*

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

Wut about cantor's ordinal?

Omega acting all gangsta until Epsilon arrives.

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

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

Or just the crossover of Numberphile and Mr Beast.

@@sinom how bout ∞?

What if we planted TREE(g64) trees?

Ah but then man would never have learned how to fly, rocketed to the moon and back or explored the ocean's deepest expanses (in a safe submersible!). All of these were said to be impossible throughout history. As a species gifted with reason (though not always accepted) we must stride boldly towards new humility for humility is always the beginning of knowledge! That said, these math guys are just plain out of their gourds here!

@@joeblog2672safe submersible..

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

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😂😂

@@R3cce sound fun , link pls.

@@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.

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

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

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

that's a lovely one

unfortunately subtraction isn't defined for infinite cardinals

Infinity (aleph null) minus one is infinity

Klein bottles?

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

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

@@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.

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

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

Hey it’s me you stole my comment cool idc

Germaphobe I don’t care tho

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

Bazillion + 1.

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

....2 bazillion plus infinity🙀🙀🙀🙀

@@xexpo 2 bazillion-fantastillion

I remember it being "uncountable"

Top TREE

Ironic that they're called "ordinals"

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

Gamma gamma zero (;

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

@@j.hawkins8779 Add 1

@Nix Growham

It's not even his final form!!

It's over 9,000!

@@omri9325 WHAT 9000?!

@@omri9325 I mean, you are technically correct.

Mathematicians after creating the number galleohalivitoxipityisnlotopiscisis22: 😮

ITS TIME TO STOP

@@jolez_4869 laughed too hard at that

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

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

Don't go into the TREES stop stop.

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

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.

BACHOMP There probably isnt enough quarks to reach that number

@@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.

that factorial at the end

Exactly 👌🏻.

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

@@caringheart34 I thought the same thing 🎯.

@@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😂

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.

Newton/Leibniz be like this when inventing calculus.

A problem when looking at my account balance

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

Sounds like a racist statement :(

The only finite thing that's a problem is the finite nature of human intelligence.

And yet they didn't get to Aleph-one

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

Ordinal what?

Well they didn't even talk about fundamental sequences

It's all about the juice

LOLOLO

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

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

it's rigorous enough for me!

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

@@DFPercush Exactly 👌🏻🎯😅.

@@bigpopakap Same here 😌.

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

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

Yea that inspired me to watch this numberphile video.

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

@@billvolk4236 dude, what is your problem

I agree, but I understand why they don't.

Its pretty easy for folk like me with an IQ of 80 so these folks with IQ nearly fifty percent higher can understand these numbers and the growth rate by which numbers are made. That is true but the FGH they mention in this video is like addition compared to the highest ordinal they mentioned ok said video. This process goes on for infinity. So absolutely infinity can't exist since there is more than an infinite amount of such.

and so onnnn

P O T A T O

Yep. That joke's got layers, man.

Ummm... Not true....

It's the one thing here I can comprehend

@@pleasuretokill same

The jingle on it keeps me living

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

NEVER

no

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

No

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.

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

@@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).

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

@@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) :-)

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.

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.

@@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.

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

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

its on wikipedia

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

There's also Giroux Studios

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

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.

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

lol

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

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

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

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

Entirely logical. They need many MANY moar trees for all their paper. So they ask people to fund more trees. Makes eminent sense to me!

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

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

Google User, Yes.🙂

MILDLY INTERESTING

Thats \varepsilon

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.

@@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.

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

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

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.

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

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

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

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

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

@@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.

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

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).

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

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

theres no proof it's between those@@R3cce

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

TREE(TREE(3)) joins the game

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

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!!”

@@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.

@@isaacwebb7918 Wow damn. Thats interesting!

_”I can count three trees!”_

@Lakshya Gadhwal lol

But his video was about cradinals

same lol

Was reminded about aleph

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.

iUFOm, Same for me too.😊

That's beautiful.

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

NoriMori, I’ve watched it yes.🙂