Numbers too big to imagine 

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Nice ur a googology fan, been working on many projects for 7 yrs now!

Your good im happy when i get a b on my algebra honors two tests

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Yeah infinity is just that big lol

The idea of generating a completely random positive integer seems bizarre to me, because no matter what result you get there should be a 100% chance that the number generated should've been bigger, since there are always infinitely many more integers larger than it but there must be finitely many smaller than it (otherwise you don't have an integer; all integers are finite). If you generate 3 such random numbers, does each have to be bigger than the last? It should be a 100% chance right? What if you look at the third number first, and then look at the second number you generated? There should now be a 100% chance that it is bigger than the third... I don't think the concept itself makes sense.

to “pick” a random positive integer it needs to first exist. The irony of this is functionally speaking the chance that its bigger than graham’s number is ZERO

@@cc1drt The probability of an action resulting in a certain outcome being 0 also requires the action to be completable. So really the probability is not 0, but "NA" (Not Applicable - as in the question can't be answered)

@@randomaccount2448 If it were at all possible, you would be guaranteed to pick an integer, because you only have integers to pick from. You can't pick something that isn't in the set.

YEAH IKR

A perfect example of, "Boy, that escalated quickly."

And tree 4 is your weight in tonnes

Imagine tree 4

I didn't understand that tree number 🧐

lmao

What about Tree 2.5: ?

​@@dough9512undefined

41 seconds in got huge numbers

A matemática é incrível 😍

Shower thoughts be like:

actually, Infinity is NOT a number you can be either closer or further away from. Infinity is a concept for something that doesn't HAS NO END. something without limit. The real numbers are infinite, because they never end.

@@PraiseChristTheGodtrue but they are the smallest infinity. And there are infifnities much bigger.

Fax

Personally I prefer TREE(3), since it's based on relatively simple rules that are able to bloom into such a big number without touching the infinity.

​​@@Vhite my issue with TREE(3) is that you can say it's larger than Graham's number, but there isn't really an easy way to show it, so the default answer is "believe me bro".

@@alexandertaylor7316 Well, THREE(3) is demonstrable, but you basically need a math PhD... So it is indeed "believe me bro" for at least 99.999% of everybody. On another topic, I dare say, mathematicians overthink waaaaay too much...

@@user-je3sk8cj6g 3(3)

How do we know that

​@@sylencemouse1860 well every power of 3 ends in 1, 3, 9 or 7 starting at the zeroth power. So as long as you can show that Graham's number is 3^(4n+3) or 3^(4n-1) then you know it ends in 7 Now I don't understand Graham's Number well enough to show that, but presumably, that is how it would work

much appreciated!! @@johnhawkins5314

@@johnhawkins5314 TREE^g63(g63) where the exponent acts like it does in sin^

​@johnhawkins5314 I have a similar theory. Well stated. Basically, math, patterns, observe and compare said pattern to which "power of 3 digit" each of the earlier phases of G would land on. Then yeh......??

a true scientific mind! Don't take things for granted, proof is required :)

Wow! I really respect the dedication 🫡

Nice. Keep going.

I still don't quite understand the rules on how TREE works. What does "not embedded in previous tree" mean exactly?

​​@@thesenate1844you cannot steal the tree

1:Your Mom's Weight

What about g3^^^^^^3?

And when you figure that any upcoming number is practically so much bigger than the previous one, that it's just ridiculous!

Beware this vid can giv u a numberphobia

look up "busy beaver function"

g0 is insane

because you don't imagine infinity, you imagine something that doesn't end It's close but not the same, it help to understand what it is but you don't imagine it Anyway human brain is bad with big number. And it doesn't have to be this big before the brain goes "yeaaaah something like that maybe, doesn't matter when it's this big" Just imagining a 20km thing is hard as heck. You can try to picture it next to thing that size but it's already to a point where the only thing we could compare to are pictures made from the sky And it's downright impossible to understand how big are the earth, the sun or the solar system. Very small number aren't easier tbh

A google to the googleth power. Infinity as it will take beyond the heat death of the universe to calculate those numbers.

This makes no sense; infinity is not 'imagining something going on forever.' First off, you cannot imagine that, because all you are doing is imagining something going, then ceasing to imagine that, so you haven't gotten anywhere close to imagining forever, and lastly, infinity is an infinitely large entity, not a 'process that keeps going.' So you are so terrible at imagining infinity that you have fooled yourself into thinking you could more easily imagine infinity than a really large number, which only speaks to the fact that imagining infinity is far harder than imagining any finite number, no matter how large.

​@@pyropulseIXXIfound a pseudointellectual! infinity is definitely easier to imagine than tree(3). infinity is easy, it's infinity, and basic logic that we take for granted stops working there. everyone knows that, simple. but with numbers like tree(3) there isn't anything fundamentally different bewtween them and say, 31. they're both just positive integers. but the scale pf tree(3) is so unimagineably massive, that it becomes easier to think about it as just being "basically infinity" dispite having much more in common with integers that we use every day than with infinity. and that there's the rub. we think of tree(3) as being equivilent to infinity, because that concept is easier to comprehend than tree(3)'s true size.

Hello there! I think the reason Infinity is easy to understand, is down to the basic understanding we have on the concept of Infinity. We may know it as "never ending", but once you start building up your foundation from there, contradictions start appearing everywhere. But then you realise the exact same thing can be said for TREE(3) or g(64). In conclusion; we might have a better understanding of these large numbers than Infinity. I hope you can see my view, and thanks for reading!

lol says you, at 1:50 my brain turned off and i didn't catch anything past that

Geometry Dash reference?!

*what do I expect*

didnt think id see you here

@@megubin9449 we seem to all be getting recommended the same underrated math channel

It would be easier to just say it felt like a 3blue1brown video.

@@qwertek8413 yeah, but that wasn’t the first thing I thought of

Why do RU-vid views freeze at 301?

Yes lol, I was wondering about this too

The pinned comment already says about that

Exactly

I wonder how big it could be or such just end at 6 inchs

Hexation Hexation Octation​@@Raj10896

underrated comment 🤣

Mathmatitions don't know about that one

after that, there is migration

I see what you did there😎

Cant even comprehend level 0

​@@Mountain_2Gotta be in 4th grade.

That's why I hate it when people so recklessly use infinity as a number to count with. Infinity is way bigger than any of these numbers. Infinitely bigger. In fact, tree(3) n-ated by tree(3) where n=tree(3) would still be infinitely smaller than infinity. Which is why it's pointless. They say "infinity+1 is bigger". I say it's not, infinity already contains infinity+1 and infinity+infinity and infinity power infinity, and tree(infinity). It's not limited with any finite answer so assuming anything may be bigger is just illogical. But it's easy to imagine. A perfect mathematical circle has infinite sides. All possible trees in the palm of your hand.

​@@RedGallardoThe more you think about, the less infinity seems like a number and more like some incomprehensible eldritch horror from another dimension.

Please do a video over your fascinating comment.

doesn't tree use buchholz's ordinal?

@@0x6a09 i thought it used ackerman's ordinal..

@@MyOneFiftiethOfADollar I recommend this series: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-LsQR2gHQYUc.html Specifically Part 9 goes over the Veblen ordinals

@@0x6a09 Wikipedia says that the small Veblen ordinal is used, on both the page about Krustal's tree theorem (the reason why TREE exists as a function) and on the page about the small Veblen ordinal.

The problem with trying to explain it is that the explanation itself requires a much deeper understanding of mathematics than it seems. I'll go on a -slightly- pedantic rant and then try a metaphor to explain it anyway, and apologies if at some point this comes across as condescending. It's not, I'm just trying to _really_ make it as simple as possible. Apologies also to whoever this oversimplification might offend. To most people, mathematics is just another science subject out there, but the reality is that it goes so deep and is so vast as to, in my opinion, be larger than all the other subjects (physics, chemistry, engineering...) combined. The mathematics taught at highschool level feels comparable to learning to say Ni hao, which is "hello" in Mandarin and Cantonese, and calling that being fluent in all the Chinese dialects. A lot of the proofs out there, even for things that seem like they should be "easy" to talk about, require a completely different dialect of mathematics to talk about. You need to peel it back to the abstract logic and go from there. An example of one such dialect (first order logic) would be the following sentence: ∀x ∀y ∀z x ≤ y∧y ≤ z → x ≤ z (for every X, every Y, every Z such that X is smaller than Y and Y is smaller than Z, it follows that X must be smaller than Z). It expands the concrete analysis of, say, an equation, to an abstract observation about variables without worrying about what those variables actually are. For this specific problem of the TREE function, we need to take another step back into second order arithmetic, which is used to further expand and talk about some properties and relations between mathematical objects. For instance, the sentence ∀P ∀x ( Px ∨ ¬Px ) would fall under this category (for every formula P and every variable X, either that formula with that variable is true, or _not_ that formula with that variable is true). It is within this dialect of mathematics speaking about properties of objects that we can construct a proof both that the TREE function is finite for any finite values passed to it and that TREE(3) is much, much larger than Graham's number. Rant and semi-formal explanation over, I'll put it in software terms, which bears striking resemblance to mathematics on many levels but is much easier to grasp: Picture a random mechanic in a random videogame that you can toy around with, familiarize yourself with and learn to use (can be something as simple as jumping). But to know _how_ and _why_ it works the way it does beyond "press this button and it jumps", you need to learn the programming language it's coded in, and go dive into the code. And then you might realize that just from the code you don't fully grasp how it does what it does, and you need to actually _learn how the programming language itself is built_ and go almost all the way down to how the machine functions at a physical level in order to know how the actual code works, and only then fully understand the mechanic. TREE(3) is one of these mechanics, it's concept is very simple, but to actually know how and why it works the way it does you need not only to look at the code, but know how the programming language it's coded in works itself. Those would be first and second order arithmetic, whilst playing the game is just regular math.

@@Z3nt4 hmm fair enough. I'm confident I can understand a real explanation, but if it would be exceptionally long winded and too hard for most, that might explain why he didn't put it in. I can read your statement of the transitive property by the way. :-) do you know of a video or paper that explains it properly?

@@mike1024. A proper explanation (which I'm not privy to) requires some deeper undestanding of graph theory, in which I am no expert and don't necessarily know of any readily available resources on the topic. However, if you're set on going down the rabbithole I guess you could start by looking up Kruskal's tree theorem and working your way back from there (which is NOT trivial by any means). The massive TL;DR is that under graph theory you can prove that any tree (the mathematical object 'tree') of the same type as the ones built through the TREE function must be finite. How one would go about proving that in the first place is beyond me, but that's the tool for the job.

@@Z3nt4 I'll play around with it! I've taken a couple of graph theory classes and seen some tree based proofs. Thank you.

Here's a way to put it in scale, brak an atom in half and get a hydrogen quark, an unbelievably small substance, fill the entire observable universe with those quarks and were about 0.0000000000000000000000000000000000000000001% of grahams number, lets shrink this quark filled universe to the size of a quark, then fill the universe up with it, repeat this roughly a million times and chances are, your number is still smaller the tree(3) by ALOT, when i say alot, I mean you can divide tree(3) by the amount of atoms in this universe and itll still be higher than the extremely densely packed universe This probably didn't help

TREE(TREE(TREE(3))

After TREE(3,600,000), We would run out of humanly distinguishable colors. After TREE(16,777,000), We would run out of RGB 32-bit colors.

The limited resources of this universe can not accomodate a representation of this number... But, although colors in the visible spectrum are finite, there may be no ceiling to how much energy a photon can pack... Neither a lower limit on how low the photon's frequency is possible. So, whether we'll run out of colors is questionable, we would run out of energy faster. P.S: if you want to destroy the universe, task an AI singularity with calculating every TREE(TREE(3)) tree. Tell it not to stop until it got the answer.

@@MatthewConnellan-xc3ojSSCG(3.6m) perhaps?

​@@MatthewConnellan-xc3ojAfter TREE(TREE(TREE(....... we would run out of TREES cause we used too much paper to write them on papers

if you understand math in the first place, that is

This video is severly underated

​@@vincentjiang6358nuh uh not the video only the guy who made it is also underrated

but only finitely more interesting. Maybe around the factor TREE(TREE(3))

math is more interesting on its own; what you just admitted is that you are not interesting and need someone else to program your mind with ideas that are interesting on their own. This guy is not making math more interesting; he is literally just talking about the math, and the math is interesting on its own. I am amazed at people such as yourself

That's because doing something like (a^m^n) using the power law means you're just bundling it into a single exponentiation term. With tetration of a number, you have to start at the top of the tower and work your way down - that's how the larger numbers are built as you're defining a new concept/sequence. So 3^3^3^3 = 3^(3^(3^3)), noting the brackets to determine order. This then works out to be 3^(3^27), or 3^(7.62x10^12), or three to the power of 7.62 trillion.

@@TheSpotify95 Thanks.

What is the card game that has that humongous amount of possibilities?

Please share the question

Thirded, would love to see the question!

fourthed

graham's numbered

i think the same

The finiteness follows from Kruskal's tree theorem. It's not something that can be explained in a youtube comments section.

@@yxx_chris_xxy I figured it was something quite complex, but all of the youtube videos I see on it have dumbed it down too much. Maybe you could do a video at least explaining Kruskail's tree theorem, in simplified terms, and at least allude to the techniques used to compare two gargantuan numbers like g(64) and TREE(3).

Graham's Number is G64 not G63. G1 is 3^^^^3 not G0.

nice BTC pfp

E:NN(x) is x^^^^^…x with x+1 up arrows. I just thought of it.

And then you can just go on with E:NN(E:NN(E:NN(E:NN… to insane lengths.

Yeah I think the largest numbers used some technique where they turned infinite numbers into mind bogglingly big non-infinite numbers somehow

@@MatthewConnellan-xc3oj E:NN(x) = x^...x+1...x E:NM(x) = E:NN(E:NN...x)))...) with x nestations of the E:NN function onto x E:NM2(x,y) = same nestation on E:NM, with y nestations.

I remember that thread from XKCD, it was an epic thread with hundreds of posts. After studying the math for quite a while I was able to understand most of it. That thread got me deeply into large numbers for a couple of years. I made about fifty pages of notes on large numbers, including a couple dozen of the numbers listed on that thread.

Damn

Good one

How does "penetration" work?

All log, no pi.

​@@JonCombolmao

x! grows slower than x^x which doesn't even come close to tetration. The levels of recursion required to represent a number as large as Graham's number (let alone TREE(3)) go well beyond factorial.

​@@samcertified7178and (x!)! Grows unfathomably fast... 1!! = 1! = 1 2!!, same thing 3!! Though... 3!! =6! 6! = 720

Factorial world be great great great great grandkid when compared to those Pappas

I struggle to get my head both around that and also hoe if tree 3 is so big tree 4 or tree 1000 are still finite.

@@gareth2736eventually you will run out of trees because of previous trees

I had come to terms with infinity, that there is no end. I had not come to terms with how insanely large finity could be!!

When I was little I also thought things like that, about how we are the only thing that exists. There is nothing after death there is no way to escape since this is the only thing of the only thing. I was a weird 8 year old kid.

same. I think there should be maths appreciation at school where we get taught cool shit about what maths can do but don't actually have to do any sums. like I can appreciate sports without having to jog you know!?

Seems like you found a good way to imagine infinity, if it’s giving you the creeps.

how stupid; there is no finite amount of real numbers between 0 and 1, so this is utterly obvious and not creepy at all

​@@pyropulseIXXI< I have to be an asshole on the Internet for no reason whatsoever.

Good thing that maths is a close imitation but fundamentally an imitation of reality irrespective of it's unreasonable utility in bits and bobs and things that make you go hmmmm. 😊

But maybe yeah I'm more than likely wrong maybe maths is the only thing that's real and it's reality that's the charade

Why im seeing ya in all vids now LOL

hi mathis, found ya

Mathis! Of course we can keep on going after omegafinruom right?

It’s best to think of uncomputable numbers as diagonalizing over the process of creating functions itself.

its not exactly the largest number, but it is the largest well-defined number.

I just made a larger number: Large Number Garden Number + 1

@@ieatcarsyum8248hahaha large garden number+2

What's that? I don't study complex math, so I've never heard of it.

Yes, they should be ">". Sorry for the mistake.

@@digitalgenius111 you did it again XD

@@digitalgenius111

​@@ChrisMMaster0Chat-GPT be like

Sure! As I mentioned earlier, Graham’s number G63 is equal to 3 ↑↑↑… (with 63 arrows). To express this number in scientific notation, we can use the following steps: Convert the number to decimal notation by writing it as a power tower of 3’s: 3 ↑↑↑... (with 63 arrows) = 3^(3^(3^(3^(3^(... (with 63 threes) ... ))))) Count the number of threes in the power tower. In this case, there are 63 threes. Subtract 1 from the number of threes to get the exponent of the scientific notation. In this case, the exponent is 62. Write the significand or mantissa by dividing the original number by 3 raised to the power of the exponent: 3 ↑↑↑... (with 63 arrows) / (3^62) = 1.611... × 10^19728 Therefore, Graham’s number G63 expressed in scientific notation is approximately 1.611 × 10^19728. I hope this helps! Let me know if you have any other questions.

it took you a year to not learn, yet keep asking, a question that a literal 6 year old could figure out on their own in less than 12 seconds

@@pyropulseIXXI I discovered power towers on my own, essentially tetration and I learned of Graham's number, but didn't understand it. I never knew it was actually called tetration until now, nor the official notation. But you sir, have had a difficult day, to be sure. I am deeply sorry for any stress in your life, and I want you to know that there are so many people who love you; and they still love you, even if you write snarky comments on RU-vid.

disagree, you should just put the number 5

Infinity isnt a number

@@stone5401 i didn’t say infinity is a number

​@@stone5401Neither is 0

such inefficient notation

@@pyropulseIXXI I know

Dang bro

ah, yes, i know some of those words.

Sir, this is a youtube comment section.

People on the internet are not going to understand bro don't bother explaining FGH to them.

You know all that, but you don't know how to use paragraphs

Since he started it off with 3↑↑↑↑3 as G0 it makes sense in this video If he had started off with 3↑↑↑↑3 as G1 then it would be G64 instead of G63

it's g64 if you define 3^^^^3 as g1, but in this video it's defined as g0

The only difference in this video was that g1 (hexation) was defined here as g0, and g2 (the thing with so many arrows we can't write it down) was defined as g1. The principles are still the same.

@@TheSpotify95 - The principle's the same, but it still matters if it's literally wrong, especially THAT wrong. That's like saying that a google is 1 followed by 99 zeroes. Sure, you're close conceptually, and may still make your point, but you're literally giving the wrong definition of something with a very clearly stated and well-known definition, and you would fail by putting that answer on any test.

​@@rodjacksonxGoogol. Be precise when criticizing others about precision. 😉

@@Instructor876 - If you find that comparable enough to be worth mentioning, more power to you.

The same thing but negative

Well, what is a large number?

It totally depends upon the person.

@@datguy3333 problem with that reasoning is -1 > than all other negative integers

I\♾️

they are

There's two things: First off, an entire tree must be contained. Having a part of a previous tree contained is not the issue so just because one tree contains two connected blue seeds doesn't mean any future tree can't contain two connected blue seeds. Second, it is about the NEAREST common ancestor. Yes any pair of blue seeds in the fourth example have a green seed as an ancestor. But that wouldn't be the nearest one, instead it would be the lower one of the two blue seeds themself as that one is already an ancestor to the other one. This might be a bit hard to put into text form for a YT comment and for a relatively short video it would of course be difficult to go into a lot of detail on this. Though if you want Numberphile went into some more detail during their video on Tree(3).

Rayo's number is like saying "the biggest number you can define on a big piece of paper", but nobody knows how it would actually be written. And all of the other "big" numbers like TREE(3) can be defined on a much smaller piece of paper. So yeah, it's a bit of a cop out.

​@@tom-lord Graham's number was actually used in a proof as upper limit, of course you can always build bigger towers, use more arrows etc. There's some short computer program that prints out a number larger than TREE(3) but smaller than Rayo's, and the cool part is how to prove it actually stops.

@@magicmulder I meant Rayo's number. Typo, sorry.

Yes they are, they should be ">". Sorry for the mistake.

All good love the video

​@@digitalgenius111You did the mistake again lol

@@digitalgenius111

​@@ze_kangz932😂