The Largest Numbers Ever Discovered // The Bizarre World of Googology

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What is the biggest number in the universe? What is the biggest number that any human has ever conceived of, or used in a proof, or been able to write down? In this video we are going to explore the fascinating world of googology, the study of unbelievably big numbers. We will begin near the bottom with "tiny" number like a googol (10^100) and a googolplex (10^googol) before realizing we will really need to do introduce better notation. We will then see Knuth's up-arrow notation that let's us form power towers of numbers, repeated exponentiation called tetration that is much like how exponentiation is repeated multiplication and multiplication repeated addition. And it goes up and up from there! We will see Graham's number which infamously got the googology ball rolling when it was used as an upper bound for a problem in graph theory and finally we will the unimaginable Tree(3) which comes out of a simple to state problem in graph theory and amazingly results in this incredible number.
You can check our Ron Graham himself introduce how Graham's number comes about over on Numerphile here: • What is Graham's Numbe... . You might also like to check out the googology wiki here: googology.wikia.org/wiki/Goog...
0:00 Intro to Googology
1:13 Googol and Googolplex
2:40 Towers of Exponents
3:58 Knuth's up-arrow notation
9:00 Graham's Number
11:21 Tree(3)
18:18 Brilliant.org/treforbazett
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[@DrTrefor]

Or should I have titled it "largest numbers ever INVENTED" #mathcontroversies:D My thanks again to Brilliant for sponsoring this video ► brilliant.org/TreforBazett/

[@vapourmile]

I think you could have added when beginning with exponentiation that if the grains of sand required to fill the universe is 10^90 then a googol grains of sand isn't just a bit more sand than can fill the universe, it's enough sand to fill 10 billion universes.

[@tetraedri_1834]

Tree(3) is not largest number ever invented, not even close. As far as I know, the record goes to Rayo's number. From Wikipedia: "The smallest number bigger than any finite number named by an expression in the language of set theory with a googol symbols or less". And almost by definition, in some sense this class of numbers is the fastest growing sequence of numbers that is well defined in the language of set theory. Edit: @Angel Mendez-Rivera below have mentioned that Rayo isn't the largest, and the record goes to the Large Garden Number. To my understanding, it beats Rayo by using a more powerful language than set theory. You learn something new every day!

[@mathisehrhart4207]

@@tetraedri_1834 absolutely, there exist a few bigger number but they are ill defined or just a simple extension of rayo's

[@MasterHigure]

@@tetraedri_1834 I wanted to say "Busy Beaver", but that's more or less the same thing. Basically "The largest number we can describe with a given amount of real-estate and a fixed language powerful enough to describe any concrete algorithm."

[@robertveith6383]

*@ Dr. Trefor Bazett* -- Maybe you should have titled it "Largest numbers to have been given their own names."

[@davidcarter8012]

g(65) makes Graham's number trivially small by comparison.

[@conservaliberaltarian2753]

That is the amazing aspect about it. Merely by increasing the integer after G, it takes the previous mathematical answer and makes it the number or arrows in the current number. You think that's something? Compare G64 to GG1. G64 is the G sequence iterated 64 times. GG1 is the G sequence iterated G1 times. The number of iterations itself is 3^^^3 layers of power towers.

[@emuccino]

g(65) + 1

[@MagruderSpoots]

But Grahams number was used to solve a problem. That's why it's not just a big number.

[@Skywalker2016GD]

g(g(g(g(g(g(g(g(g(g(64)))))))))) sounds pretty big to me

[@TheDuckMan2523]

Tree(g(65))?

[@kingpatty4628]

As a wise man once said “no matter how enormous the number you can think of, it still closer to zero than infinity.”

[@dmntomfg]

But rayo number is closer to the infinity than to the zero :) and that is still finite :)

[@modr0160]

@@dmntomfg Rayo number in insignificant compared to infinty

[@Shoomer1988]

Infinity isn't a number. You might as well say it's still closer to zero than a bowl of petunias.

[@itzashham797]

​@@dmntomfgits not

[@asheep7797]

⁠@@dmntomfgnah its further from 3 times Rayo's number than 0.

[@dayisnow]

Took Discrete Math with you at UC in 2019. Awesome to see your channel blow up. Only math class I ever got an 'A' in lol.

[@DrTrefor]

hey that's awesome!

[@axbs4863]

Wholesome

[@ravenptl]

Awesome that the guy/gal only hand an A? Mean.

[@l.w.paradis2108]

What's UC? I'm wondering whether U of California or U of Chicago, thinking probably Chicago.

[@Grassmpl]

Yes his channel blows up. I wonder what his affine charts look like.

[@conservaliberaltarian2753]

In the fast growing hierarchy, Graham's number uses the 1st ordinal (omega). It falls between f-omega+1(63) and f-omega+1(64). TREE(3) uses the 6th or 7th ordinal.

[@Xnoob545]

What do you mean 6rh or 7th ordinal it's way past psi or phi I mix them up of 1,0,0 Which is gamma_0. It's the omegath ordinal So TREE(3) uses an ordinal past the infinity-th one

[@conservaliberaltarian2753]

@@Xnoob545. That is what 'Carsbrickscity' said that TREE(3) is the 6th or 7th ordinal. He is a mathematician KZbin channel.

[@lumi2030]

TREE function corresponds to a growth rate of ψ_0(Ω^(Ω^ω×ω)) in the fast growing hierarchy (with respect to buchholz's ψ function)

[@Assymetry]

@@conservaliberaltarian2753 do you have a link to the video? In the FGH tree(3) is around small Veblen ordinal level of power, which is much, much larger

[@R3cce]

@@AssymetryYes, you are right. Actually much faster than the Small veblen Ordinal, but slower than the Large veblen ordinal. So it is between SVO and LVO.

[@nchiley]

I think any discussion of Graham's number should include the lower bound as well. The answer to the question they are trying to answer is somewhere between 11 and Graham's number.

[@helenkeller9182]

Currently it was narrowed down to "between 13 and Graham's number" :) Mathematicians making their way slowly but surely

[@Arthur-io4ey]

@@helenkeller9182 The upper bound has also been reduced since a lot of time, it's not G. Currently an upper bound is 2 ^ ^ ^ 6.

[@jakerussell135]

i thought the upper bound was 2^^2^^2^^9 now? and yeah the lower bound is still 13

[@amits4744]

@@jakerussell135 heard that it's down to 2 ^^ 2 ^^ 5138 now

[@smoceany9478]

it would be so funny if it turned out it actually was just 13

[@vincentvandergoes444]

A googleplex does have a physical meaning. It is the type of timescale where you will start to observe significant failures of the second law of thermodynamics. Entropy doesn't *always* increase, it *almost always* increases. In a googleplex seconds / planck times / years (pick your unit, it doesn't matter much), you might see a boltzman brain spontaneously forming.

[@samueljehanno]

Wow

[@fordid42]

Yep, and around 10^10^120 years the universe will reach a state of thermal equilibrium. Nothing will ever happen again except for quantum fluctuations which can cause Boltzmann Brains to appear around that time, and on an even vaster timescale (10^10^10^56 days/years/whatever... won't make any real difference here, either) a new Big Bang event could occur.

[@richardkeck4918]

You could say the same thing about any big number

[@danielrhouck]

3:22 Unless I miscounted, the previous number was *much* larger than this one. Sure, googolplex is unimaginably greater than 10, but it also takes more screen real estate to write, and the extra 10s you could fit in more than made up for that.

[@guillaumelagueyte1019]

I thought the same thing, and it made me think of the large number competition when the competitors wrote on a board, and one of them replaced the 999999999... by 11111111... because it's possible to squeeze more 1s than you can squeeze 9s in a given amount of writing space

[@BrazilianImperialist]

Bruh, this is a googoltriplex, it is much largar than the previous

[@danielrhouck]

@@BrazilianImperialist A googolplex is 10^10^10^2 so this is 10^10^10^2^10^10^10^2^10^10^10^2^10^10^10^2. Thatʼs less than 10↑↑16. The previous screen has 10↑↑20.

[@BrazilianImperialist]

@@danielrhouck No, it is 10 arrow arrow arrow 10

[@bgmarshall]

@@BrazilianImperialist no it isn't

[@loganm2924]

As someone who kind of abandoned the finite numbers in googology in favour of infinite ones which I found much more interesting. I’d love to see a video on transfinite ordinals and cardinals :)

[@Assymetry]

Agreed

[@scubasteve6175]

woah what are those lol i'm not on that level yet clearly

[@grox2417]

@@scubasteve6175 it's not really about your "level", just a mathematical curiosity. You can check out a video made by Vsauce to get more than enough info: kzbin.info/www/bejne/iaO4aox6pL14bpo

[@0x6a09]

@@scubasteve6175 I think you should try to understand what "fast growing hierarchy" is, it is a simple functions that use transfinite ordinals to create very strong functions. It probably can describe numbers bigger than TREE(3).

[@egwenealvereiscool7726]

@@0x6a09 Yes - Since you can define as many infinite ordinals as you want, they define Gamma nought to be faster than all of those using diagonalization (like the jump from finite ordinals to omega). TREE(n) is on the order of Gamma0(n). its crazy that it literally takes 2 infinite layers (the finite ordinals and infinite ordinals) to reach a function that grows on the order of TREE(n)

[@adb012]

"There is no largest finite number" Plot twist: There is no largest infinite number either. There are infinite sizes of different infinities.

[@nsinkov]

"infinite sizes" Sure, but which size infinity describes the number of sizes of infinities? ;)

[@adb012]

@@nsinkov .... I suppose that Aleph Null. Put the smallest infinity, then the second smallest, then the 3rd and so on. You can pair them with the natural numbers, except....

[@elenplays]

Since we're mostly going by Cantor's rules here, there is a largest infinite in capital omega Ω, appropriately named Absolute Infinite. It's the set of all ordinals, including infinite ones, so ω is a subset of it, and so are all the other infinities.

[@TotallyTaliton]

@@elenplaysthe infinity of infinities. If we could count to infinity, we would have to do that absolute infinite times, each one getting harder to count to until absolute infinity difficulty level.

[@yablomas_]

@@nsinkovabsolute infinity

[@youregonnaletityeetyouaway2882]

i used to be really into googology. tbh i came for the ridiculous names and stayed for the interesting maths. id love to see a video on busy beaver or BEAF

[@bryantofsomething5964]

Oh yes! I would adore a video on BEAF!

[@spelpotatis7240]

I would love to see some more videos on this. As a googologist myself, i'd like to say that it would also be worth it to check out a bit about ordinals, as that's where the true googlogy comes in. You could discuss things like the fast grwoing hierarchy (Which converts transfinite ordinals to finite numbers), ordinal collapsing functions and stuff like that (When it comes to way to produce ordinals, again, i would recommend ordinal collapsing functions, but something called bashicu maatrix system would also be really fun to see a video about, as it's a really simple way to make extremely large transfinite numbers.) It could maybe even be fun if you could make your own little googology series where you discuss numbers that get lrger and larger each episode, but i understand if you don't do it, because it is kind of a niche subject

[@Assymetry]

Oh hello spel

[@Xnoob545]

Oh hello spel

[@MustafaAlmosawi]

There’s a great Numberphile video where the fast growing hierarchies are used to compare Graham’s Number and Tree(3)

[@Cessated]

i like googology but i'm still terrible at it also hi spel

[@StoicTheGeek]

I’d be interested in videos on Conway chain notation and Loader’s number

[@Memer9456]

it may look like im crying but that is just my brain melting through my eye sockets

[@TIO540S1]

Since a random integer chosen from “all integers” has a probability of 0 of being smaller than any number you’ve defined or any number that any one ever has defined or ever will define, I contend that all defined numbers are negligibly small.

[@FireyDeath4]

I mean...eventually life will come up with a large googologism they don't think/happen to surpass before it's extinguished from the universe

[@fullfungo]

Cool idea, except there is no notion of “a random integer” if you want the distribution to be uniform.

[@TIO540S1]

@@fullfungo yes, there’s the rub all right.

[@michalmaixner3318]

@@fullfungo well you can always reformulate the idea to "for every k you can always find n such that probability of choosing number smaller then k from the interval (0,n) is negligibly small" which would make the statement "I contend that all defined numbers are negligibly small" sensible.

[@TIO540S1]

@@michalmaixner3318 I will have to construct a mathematically valid argument that captures the idea. An idea we all understand, by the way.

[@aqwaa3057]

the worse the audio quality, the better the video. Great work man!!

[@yanisbowrin4321]

For Knuths up arrow notation, remember you can always also use the "^" symbol. E.g. 10^10^10^10^10^10^10^10^10^10 = 10^^10

[@user-sy9dx5zp9d]

10^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^10

[@user-sy9dx5zp9d]

9999999999999999999999999999999999999

[@TheScientificSpot]

10^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^10 💀💀💀

[@bpsbandit1344]

Fantastic stuff, didn't expect to have energy for more math during the 3rd year grind, but youve got me hooked XD

[@DrTrefor]

Great to hear!

[@kamitebyani5309]

This is a great video . I enjoyed this video so much. Thank you for putting together all these amazing numbers and your explanation in one video. I have subscribed and can’t wait to see more videos. Thanks

[@jonnaking3054]

Discussing the googolplex with my parents had led to some intense shouting matches. Lol when I tell them that a googolplex is 1 followed by a googol zeros, they can understand how that's different from a googol. They are like " One with a googol zeros would be a googol!" Then I try to explain it to my aunt and she doesn't get it either !!

[@zjz1]

Same happened when I tried to explain moon rotate 1 time a month, not zero. That's why it always face earth with the same side, if it's zero then we can see the other side every 15 days. I even use my fists as model, but in the end I only want to put fists on their face.

[@jonnaking3054]

@@zjz1 ikr, it's frustrating bc I feel like I'm not explaining it correctly. I will say "A googol has a hundred zeros and a googolplex has a googol zeros, there's not enough space in the whole universe to write out googolplex" and my dad is like "Why!? It's just a hundred zeros!"

[@TheArtofCodeIsCool]

oww yeah dad? So following your logic, a thousand has how many zeros?

[@Xnoob545]

@@jonnaking3054 replying to both you, and the comment: explain to them that every zero you add makes the number 10 times larger

[@davisatdavis1]

@@jonnaking3054 try giving them something different for them to truly understand what you're saying. Say that a googol is 1 followed by 100 zeros, but that doesn't make it equal to 100. Something smaller like that, that they could comprehend. Or 1 followed by 3 zeros, doesn't make it equal to 3.

[@ssarmazi]

Thank you for explaining Tree(3) so well.

[@DeadJDona]

Numberphile also has a video comparing TREE(Graham) and Graham's Number of TREE (3)

[@TIO540S1]

It also touches on the fast growing hierarchy. It's an excellent video.

[@TotallyTaliton]

Tree(g64) and gtree(3)

[@magicmulder]

The first counter example to the Collatz conjecture is larger than TREE(3) but unfortunately this comment section is not wide enough for my wonderful proof.

[@brandonmtrujillo]

Numberphile is great but thanks for making these two numbers easier to understand

[@iamthestormthatisapproaching69]

Imagine being this one evil dude coming up with a bigger number than their competitor, just to increase it by 1

[@txikitofandango]

The thing that always bugs me about these notations is... okay, if I want to write a power tower of 65,536 2's, there's a simple notation for that. But the chances of me being able to notate a number that's in any neighborhood of that number are zero. There will probably never be any system of notation that could cover a range of big numbers because they contain too much information.

[@BrazilianImperialist]

There probably is

[@thestarvingonetso5627]

Not a mathematician, but I think that once you get the arrows down and you want to cover something else, you just have to use the smaller stuff like + a, + x^y or anything of this kind that eventually leads you to that number by smaller bits.

[@txikitofandango]

@@thestarvingonetso5627 that's still going to cover a tiny, tiny neighborhood of whole numbers around the big one

[@txikitofandango]

@@thestarvingonetso5627 You get what I'm saying? If you start with 3↑↑↑↑3 and try to add or subtract numbers from it, even big numbers like 3↑↑↑10000 you won't get very far. There's not enough information in the universe to even write down most numbers, no matter how clever of a notation you come up with.

[@philip8498]

well, if we want to get really technical you can write you 2^^^^2-n to write any number in that numbers vicinity. we just dont know how that number would look in in our base 10 notation.

[@mihirrao10]

Just an idea: A series/video on the Mathematical Analysis of Algorithms/Asymptotic Analysis might be interesting.

[@DrTrefor]

I like that idea!

[@rykehuss3435]

When you get into stuff like TREE(3), it really becomes more about functions and how fast they grow. This is represented in a thing called fast growing hierarchy. Numbers lose meaning at this point, and googologist are more interested in creating functions that grow faster than other functions.

[@jolkman1]

I can recommend looking into the Ackermann function, also a way to generate ridiculous numbers. And it can be understood and written using knuth‘s arrow notation

[@NicoPlayZ9002]

or fast-growing heirarchy ( fx(y) )

[@zunaidparker]

Is there a way to fix the "hollow" audio with some processing? Maybe make it mono to remove the echo effect?

[@okkoheinio5139]

I just realized a really nice fact. With Knuth's up arrow notation, f(2,2) = 4 holds for all levels. because 2 up n 2 is 2 up (n-1) 2

[@igxniisan6996]

8:21, three up arrow operation is also known as pentation, four up arrows will be known as hexation, next up septation, octation and so on....

[@wiggles7976]

For the kth graph in the tree sequence, can you choose any node in that graph to be the root node when you are testing to see if an earlier graph can be embedded in it, or is the root node of the kth tree set to be just one particular fixed node?

[@davidjericho4815]

With the Graham sequence, the output of each layer (G1, G2, G3, etc) makes the previous layer roughly equal to zero in comparison. G63 is a rounding error compared to G64.

[@MichaelDarrow-tr1mn]

g64 is not that small

[@Grassmpl]

Yeah I betcha G64/G63 is much bigger than 1/machine epsilon.

[@brycebenz9705]

"I'm not Mr. Beast." - not Mr. Beast

[@3Max]

Thanks, this was a much more easily followable description than the Numberphile video! (specifically "kth step has at most k nodes" !!). Also, I'm not color blind, but those yellow vs green were really hard to distinguish. In future videos consider making these kinds of things more easily visually distinguishable!

[@bettergaming2321]

Very good video, truly proud of you.

[@kingarth0r]

What happens when you use large cardinals in the fast growing heiarchy? (Or even just the first uncountable ordinal). I'm thinking those numbers are still smaller than rayos number because large ordinals are expressed pretty easily.

[@lumi2030]

nah, f_ω^ω should always outgrow f_ω+1 in the FGH in the same system of fundamental sequences, if they are in different ones, it might not be the case.

[@skysetblue9578]

Am a number noob but I never heard of “power tower” but I’ll never forget that lol. I love when people talk about large maths and you and Numberphile explain it so well 😊 subscribed

[@pierfrancescopeperoni]

I was thinking about this just today, what a nice coincidence.

[@onradioactivewaves]

Geez, what are the odds of that? 😈

[@robinbrowne5419]

Great video. This reminds me of when I was a kid and arguing about who had the most of something. "I have a hundred. I have a thousand. I have a million. I have a zillion." But, of course, as you say, we can always add 1 to the number. So, whatever number you can come up with in this video, I can always add 1. So, "I win". Lol :-)

[@DoomRutabaga]

This strategy is so OP that they banned it in big number competitions

[@mattkane5468]

Very interesting video. If I might make a suggestion for another video, how about one about the smallest numbers ever invented? (Smallest as in closest to but not equal to zero.)

[@NicoPlayZ9002]

perhaps 1e-(tree(rayo(gg64)))?

[@samueljehanno]

Just take the inverse function lol

[@LeoStaley]

Numbers just get so big, I like to think that there are actually an infinite number collatz conjecture violations, of looping sequences with arbitrarily large numbers of numbers which do not go back down to zero. We can just never find them.

[@gapplegames1604]

8:20 2^16 is not 16 twos in the tower, it’s four twos in the tower. great video by the way

[@chessandmathguy]

Yeah I was about to say the same thing.

[@MABfan11]

i think you should take a look at Bowers Exploding Array Function (BEAF), it's really efficient at creating ridiculously large numbers

[@Xnoob545]

And also really stupid Googologist here, hate it

[@lumi2030]

BEAF isn't that good at all

[@bryantofsomething5964]

BEAF is ill-defined after tetrational arrays, it's better to use Bird's array notation.

[@hefesan]

So long gay bowser

[@ernestezekieljoybetts2088]

best math teacher I've ever had

[@DrTrefor]

Thank you so much!

[@interest21stcentury74]

Hey dr, I hope you are doing well, I just had a glance on the game theory online course in coursera, its horrible, your game theory is way better and way more clear than theirs. I really hope you upload your version, it'll be way better. Thank You Dr and thank you for your wonderful videos.

[@weasel9815]

Now this is a long comment, but I think that it may interest some of you smarter people: I have an idea of a massive number that nobody could define, but it should not be infinite. Say that you have an infinite 3d vacuum of space and you choose a point. now you move in a random direction (up, down, in, out, forwards and backwards, nothing else like in-between any of these directions) for 1 unit of length and put a point there. you repeat this process many, many times. during a random test, how many steps could it take for you to place a point back on the initial point by chance. I mean you could just keep going and stray further and further away from the starting point and it will get less and less likely to land on the initial point with each additive step But if the step is repeated enough times, theoretically you could land back at the start, just after an inconceivable amount of steps. There is an important catch. This random path experiment takes place in a plane of existence with a much larger number of spatial dimensions (whatever that number may be, you choose. for example I choose 10^98 spatial dimensions). Of course, the number of steps in a given random experiment will drastically increase with each extra integer number of dimensions. Like a 1 dimensional random path experiment may very well be over in 10 steps (or something else because who am I to know), but a two dimensional path experiment will take many many more steps (potentially a googol steps of something else) and this pattern of insane growth would continue with each additive spatial dimension. This is my concept for a huge but undefinable number. Obviously a different number would be found for each different experiment so take that as you will. I am only in year 11 so all you smarter people in this comment section can correct bad terminology or correct this idea - feedback would be great! And this idea mainly came from this Wikipedia page and it has cool visualisations with it to help understand my attempt at a sound explanation. en.wikipedia.org/wiki/Random_walk

[@azeezabrahams9427]

and I panic when my calculus equation has a value > 10

[@TmOnlineMapper]

Wasn't there a proof that tree(n) for any integer is finite?

[@michaelkutschke6533]

These incredibly large numbers, but still coming out of computable functions, makes me realize a bit better how truly fundamentally ridiculous uncomputable functions like Busy Beaver have to be.

[@MABfan11]

Rayo's function makes the Busy Beaver look slow

[@shanetomlinson561]

i hate the audio but i love the explaination. i will now look for more of your videos.

[@disgruntledtoons]

The version of Graham's number shown here is not the one the Graham used in the proof that got all of this started. The number in the proof is F(7), where F(n) = 2^F(n-1)3 (that is, F(n-1) up arrows between 2 and 3), and F(1) is 2(^12)3. This is the projected upper bound of a solution to a particular problem in Ramsey theory, and at the time of the proof it was the largest positive number used in a published mathematical paper.

[@Philoreason]

Great content, but can you adjust your mic (or increase vol of your audio channel in your mixing), somehow the audio level is way too low, thx!

[@patrickgambill9326]

17:25 are we assuming a specific vertex is the base of the tree, or is this up to isomorphism? The two trees on the right are isomorphic

[@DrTrefor]

Yes, my understanding was the trees under consideration had roots.

[@convindix9638]

In more detail, the embedding here is that one tree's vertices form a strict subset of the other's, and ancestry ("x is on a lower level than y" for vertices x and y) is preserved

[@skalskifamily4275]

Your audio is Great !

[@bergnerm]

Good video and thanks for putting it together. One thing I like about Graham's Number is you can actually see how the number is generated. With Tree(3), you always see a lengthy discussion of the "game" on which it's based, but when it comes to proving it's bigger than, say, a googol, a googolplex, Graham's number, etc., you always just get a knowing nod, "oh, trust us, it is". That's not very compelling. And then of course, there's Rayo's number--most of the time I see that explained, they're trying to describe how big a googol is, as opposed to Rayo's number itself--which is somewhat disappointing because you can really stuff any number into the Rayo equation--it just so happens he picked a googol.

[@uridimmuvltozwta1466]

There are 10^83 particles in the universe. 10^100 is so big that if you wrote a zero on each particle you would run out of things to write on. Then Skewes number is the number that represents all possible arrangements of particles in the universe. Basically, swap one particle in two objects and that is ONE arrangement. I suppose that is the combination of 10^83? Not sure if that is bigger than the discussed numbers. Skewes number is discussed on "star talk" episode "large numbers." Great video to help really wrap your head around these big numbers.

[@DrTrefor]

Ah ye sSkewes number was one I thought about including, it is also just nuts!

[@MnMGaming69]

So skewes numbe is like 10^83!

[@khabeesschool9919]

You mean 10 to the power of 10 to the power of 100

[@fordid42]

Skewe's Number is more like 10^10^10^34. Or 10^10^10^963, depending on the Riemann's hypothesis.

[@lifesgood5065]

Love this video

[@DrTrefor]

Thank you!

[@KllswtchOvrDrv]

Great video, I wish you were my math teacher

[@Dupermirrors342]

I love your googology video

[@joshuaisemperor]

Omg I love the whole Googology thing! I hope you might cover some more about this Dr.Trefor Baaszett. Maybe you can even clear up one of my long living mystery that I can not wrap my head around. My question is: When dealing with these really really large numbers how is then determined which, if we have two very large numbers ,which number is larger than the other? This question keeps baffling me. Take for example these arbitrary large numbers: Moser's number and Grahams number. Awesome video as always Dr. thank you for making these gems!

[@DrTrefor]

It is such a cool topic!

[@mathisehrhart4207]

You can use Fast growing Hierarchies, or compare it to the same system, like beaf G(64) in FGH is f_ω+1(64) while Moser Number is, i might be wrong, f_f_5(2)(2)

[@joshuaisemperor]

@@mathisehrhart4207 Thank you for your reply :)

[@adb012]

Not only is Tree(3) humongously large but finite, but Tree(n) is finite for any n. So imagine Tree(g(64)) (the longest sequence of non-embedding trees that can be created with Graham's-number many colors). And that is just stupidly large but not even remotely close to the largest tumber that homo sapiens have come up with. Did you note that Tree(g(64)) has only 12 characters? Imagine how insanely larger it would be "the largest number that can be described using 1 googol characters". Look what I can do by adding just 1 character: Tree(g(64))! (hint, the exclamation mark is not an exclamation mark)

[@Xnoob545]

If you're talking about Rayo's Number that's a googol symbols of set theory

[@caspermadlener4191]

The factorial doesn't reallyatter here. This is much more enormous: Tree(g(99!))

[@caspermadlener4191]

Or Tree(g(9!!))

[@robertveith6383]

@ adn012 -- Tree(g(64)) has 11 characters, not 12 characters.

[@adb012]

@@robertveith6383 ... Ha! good catch, I love that you bothered to count them to find the mistake.

[@malicholson5767]

am i the only one who thinks its amazing how no matter what operation above edition is done, one remains one.

[@The-creator-of-NIBN]

Fun fact: TREE(3) > {10,100[1[1/2~2]2]2} (Dont ask me what is this)

[@paegr]

I wonder how the sizes of unfathomably large numbers are calculated, like how can it be proven that g(64) < Tree(3)?

[@angelmendez-rivera351]

There is no calculation involved. The proof is purely conceptual, and it uses sequences of functions.

[@angelmendez-rivera351]

@@Victor_StudentOfFloppa False. TREE(3) is finite, and this has been proven. It also does have a precise definition, it is just not expressible using any form of familiar notation, due to the rapid growth of TREE. It grows faster than any form of Conway chain arrows.

[@R3cce]

It does lie somewhere between the SVO and LVO ordinals in fgh

[@R3cce]

These ordinals are just too ridiculously big to understand. The SVO and LVO is beyond the feferman schutte ordinal

[@MABfan11]

@@angelmendez-rivera351 pretty sure Bird's Array Notation can reach TREE(3) though

[@user-rx6zv5en8f]

everyone:tree(3) is massive me:Tree(4)

[@eastonrocket3273]

Tree(Tree(g64))

[@eastonrocket3273]

@ZTISowner Also me: SCG(13) Random person: BB(1919)

[@eastonrocket3273]

@ZTISowner Rayo(10^10^100)

[@Jonasz314]

for more on Tree(3) vs g(64), checkout this really cool Numberphile video: kzbin.info/www/bejne/ZomcdYyIgdKHhLs

[@R3cce]

TREE(3) literally almost breaks the entire fast growing hierarchy

[@channel_._.]

@@R3cce Not even close

[@elvislopesdearaujo8644]

There was no mention of SSCG (3)... I am waiting 👍

[@etaoimusic6595]

Great video, mic quality ruined it though

[@MikeLeed]

I hope your audio production has improved since this video. I have to turn my volume all the way up to 100% to hear you talk.

[@DrTrefor]

Yup! Mic broke for this video sadly and had to use crappy back up one:(

[@iopqu]

The smallest Turing machine for computing Tree(3) would thus make more than Tree(3) steps, so the BB(x) would certainly be larger. The only problem we can't actually compute busy beavers, so we don't know how large

[@AfoteyAnnum]

This is absolutely fascinating stuff. I stumbled upon this video while looking for a meaningful explanation of Sam's Number. Thank the Algorithm Gods™ for this find~! #Subscribed

[@DrTrefor]

Glad you enjoyed it!

[@ericwelsh4853]

It's a fascinating subject, but eventually it becomes pointless. I think the example where the number of grains of sand would be greater than the volume of the observable Universe would have been a good stopping point. It's a bit like asking how large, or complex, of a concept can the human mind understand. Through abstractions it's probably infinite, I think, if you keep making new, more complex concepts built from the previous largest. Maybe the actual numbers and concepts aren't interesting now, but could be at some point in the future. So the challenge becomes constructing the new tools for constructing the large numbers -- or the tools for constructing complex concepts (and the notation) too large to fit in our minds.

[@laxxius]

It's already completely pointless from the beginning. And I think it only becomes fascinating once you pass the number of grains of sand needed to fit in the universe, *because* it stops being grounded in reality in any way. To me that is what makes it interesting: that it doesn't apply at all to anything that could ever be in real life. ...Except it may actually have a use, because some of the real big numbers rely on unsolved math problems, and have also given rise to previously unknown unsolved math problems. Even though math is pretty abstract, solving math problems has had real world benefits in the past.

[@chitlitlah]

Graham's number remains my favorite of the ridiculously large numbers. It's the only one I know that meets the following two criteria. One, it's pretty easy to explain how it works to someone who knows only basic math and make them realize how quickly it gets ridiculously large. Pretty much everyone understands that multiplication is iterative addition, powers are iterative multiplication, and can be made to understand that double up arrows are iterative powers, triple up arrows are iterative double up arrows, etc. With tree(3), understanding the problem isn't so complicated, but I'm just left to take your word for it that it's a huge finite number. I've seen no way to calculate it using steps like with Graham's number. Two, it was used seriously in a mathematical paper. The problem for which it was used isn't so easy to understand, but you can explain to someone that there was a problem in advanced mathematics for which it was proven that the answer was somewhere from 3 to Graham's number, which in itself is such an astoundingly large range, almost as amazing as the number itself. The range has been narrowed down slightly since then, but it's still anywhere from a very small number you can easily count to to an unimaginably vast number. Some of these numbers are just numbers in a sequence where it happens to get large or they're dreamed up numbers that are large for the sake of being large.

[@livedandletdie]

Graham's number was the absolute limit, to a problem about patterns in edge coloring of hypercubes, such that a simple pattern of 4 coplanar vertices all have their connected edges in the same color. What's the minimum value where a nD-hypercube will always contain such a pattern, no matter the edge coloring assuming only 2 colors can be used. It's a bit more specific than that, but Ramsey Theory is hard. And the current lower bound is 13, and the weak upper bound, is g(64), but the actual upper bound is believed to be trivially small in comparison to g(64).

[@pierrecurie]

@@livedandletdie According to wiki, the lower bound was increased to 13, while the upper bound was decreased to some mess that is smaller than g(1).

[@samueljehanno]

​@@pierrecuriebruh

[@onesoldierserving]

I've been a fan of incomprehensibly large numbers for years. I've watched Numberphile's videos on Graham's Number. I've watched Sixth Symbols' video on Tree(3). I've watched VSauce's video called Math Magic that explores 52!. I've also watched VSauce's video Counting Past Infinity. 1) NO ONE has explained arrow notation as well as you. I'm not a genius, but I'm no dolt. Something (!) about the way you explain it FINALLY made it click for me. 2) Something about the way you describe Tree(3) demonstrates the abject ABSURDITY of the number. And, you waste no time on trying to find inventive ways to "describe" the absurdity of the number. You are a lucid and effective communicator of these (and probably other) concepts and I'm glad to have found this video. Tree(G64) kudos to you! Oh! And THANK you for defining the term and giving me the name of a resource to examine even LARGER numbers!

[@colinjava8447]

It's kinda ironic, numbers like aleph0 and c and 2^c are obviously ridiculously large but I'm comfortable around them. But just thinking about a finite number like Grahams number just melts my brain as it's too big.

[@JamesTaylor-je6es]

I read if your brain could imagine Graham's Number, it would collapse into a black hole.

[@jespervalgreen6461]

Very nice. But... "there's no vocab in my vocabulary..."? Vocab means vocabulary, right? The individual entry in a vocabulary is commonly known as a 'word'.

[@DrTrefor]

Lol fair!

[@aashsyed1277]

this is your supersecret project? and at 1:36 you mean the observable universe

[@DrTrefor]

haha I have many top secret projects:D

[@bettergaming2321]

Tree 3 is enormous but imagine playing the tree game with Tree 3 number of nodes.

[@xoiyoub]

I wish someone makes a video about the proof that tree(3) is finite, and how they concluded that is that big

[@nagyandras8857]

+1 and I won. Whooooo...

[@MABfan11]

Loader's Number, Rayo's Number, Fish Number 7 and Large Number Garden Number makes all of these numbers look small

[@DrTrefor]

It's really crazy how far it is possible to go down this rabbit hole!

[@Xnoob545]

I'm pretty sure huge numbers get fuzzy because they're ill-defined

[@paillote]

Please can some video about Tree(3) explain precisely the rules, what is embedding in this case?

[@itisi2042]

From my understanding, embedding in this case means that if a tree contains any previous tree in the sequence it is not allowed. E.g. a red dot with a blue dot branching out to the left and a green dot branching out to the right is it’s own tree. This means that if another tree has a blue dot to the left and a green dot to the right, which both can be traced back to a red dot it doesn’t count as it contains that specific tree within it. This also applies even if there are other coloured dots between them, as long as it has all the dots in the same formation it counts as an embedded tree. I’m not too sure if I worded that well enough as I still have a lot to learn about this, but I hope this helped.

[@Xnoob545]

Wagch numberphile

[@alansmithee419]

3:20 This is actually vastly smaller than the previous number shown. Increasing the height of a power stack creates far larger numbers than increasing the number in the stack.

[@PrzemyslawSliwinski]

And they all start quite innocuously: 2+2 = 2*2 = 2^2 = 2^^2 = 2^^^2 = ... Am I correct?

[@abdullahimran4624]

no? 2 ^^ 2 = 2^4 = 16, and 2 ^^^ 2 = 2^16 = 65536

[@Xonatron]

You’re right. They’re all 4.

[@Xonatron]

@@abdullahimran4624 this is not correct.

[@Xnoob545]

@@abdullahimran4624 bruh moment

[@yukisetsuna1325]

@@abdullahimran4624 2^^2 = 2^2 = 4 🗿

[@jackcavanaugh6744]

How on earth did somebody prove tree3 is finite?

[@R3cce]

The most shocking thing is than TREE(n) is finite for all values of n. For example even something as crazy as TREE(TREE(3)) will be finite

[@R3cce]

It is finite because it has a growth rate limit in fast growing hierarchy. Meaning each output will give a finite result

[@opinionshurt2905]

I’m obsessed with the infinity between 0 and 1.

[@duggydo]

Good video. You should go through a proof sometime on why TREE(3) is finite and make a video. If you can accomplish that proof and make an understandable video, you would be a legend!

[@user-pb4jg2dh4w]

How can 1 googol be bigger than the univers , while the expansion of the univers is exponential

[@DrTrefor]

I mean at this point in time with the universe as large as it is today.

[@dekippiesip]

Your statement doesn't even make sense.

[@Lolbadnub]

Let’s say your mom is fat and she’s still growing, I can still say that right now a hippo is heavier than your mom, even if your mom will eventually become larger.

[@bryantofsomething5964]

dude where is your proof? we dont even know how fast the universe is expanding.

[@rv601]

Kind of poorly explained at certain points. For example at 8:22, 2 to the 16 is not equal to sixteen 2's in the tower

[@DrTrefor]

Ah true, thanks for the catch!

[@Xonatron]

I searched for this as I caught it too. I think it’s very well explained though, minus this slip. It’s a lot to juggle in one’s mind. Other videos have done t but skip over the step by step you’ve done here. I like it! So, at this point, it was comparing a power tower of 4 2’s to a power tower of 65,536 2’s.

[@Xonatron]

Just thinking it would have been cool to start to figure out was power tower of 2’s that is 65,536 in length. Just to see where we break and have to stop and show how little way was made!

[@yelenabibis9398]

Weakly compact cardinal:Am I joke to you:| rayos number:What about me.

[@paradoxreboot]

@8:00 "im not mr beast" underrated joke

[@IsYitzach]

Missed Rayo's number: The largest number that can be written with 10^100 symbols of set theory and logic.

[@tanish6035]

Infinity

[@DrTrefor]

ha!

[@pierfrancescopeperoni]

Infinity+1

[@tanish6035]

@@pierfrancescopeperoni nyc 👍😄, btw, infinity +1 = infinity.😁😎

[@pierfrancescopeperoni]

@@tanish6035 Then infinity is not a number.

[@tanish6035]

@@pierfrancescopeperoni yes, i do agree.👍

[@kevspeedruns9947]

I suggest the book ''Infinity and the mind'' by Rudy Rucker. He discusses the Ackermann series which I expected to find in this video. I never heard before of the kunth up-arrow but it seems to basically be the same thing as the Ackermann concept. Also, his Busy Beaver series probably grows faster than the Tree series as it can be proven to generate incalculable numbers (incalculable yet finite).

[@coganaj]

~5:11 I first came across the word "symbology" in "Star Ocean: Till The End Of Time" on PS2 nearly 17 years ago

[@Parasmunt]

This Graham's Number reminds me of the episode White Christmas from Black Mirror

[@bobibest89]

I think Tree(3) is the most fascinating large number. The game of trees that generates it is so simple, and it is completely unintuitive that by adding just one more seed you go from Tree(2)=3 to something far beyond comprehension.

[@vantarinitel]

I love it because it is a good solid large-but-finite proof of concept for how Feigenbaum Equations and chaos theory interact with neuron patterns. TREE(3) to me is a short rough answer to Why Can Brains Even At All

[@R3cce]

The growth rate of TREE(n) literally almost breaks the entire fast growing hierarchy for reference. It lies between the SVO and LVO in fgh. These ordinals are difficult to understand, because it is way past gamma zero in fgh

[@yanisbowrin4321]

Tip #3 After the 2nd notation, start grouping the numbers into one. This starts Bird's Array Notation. But put the number of arrows at the end. E.g. 10 {17} 10 = {10 , 10 , 17}