holy shit this is a school project????? dude this is actually phenomenal. like ive seen school projects in the past that are really good (like the one zaza made about eateot) but this is probably the best one ive seen
Yeah. Same here. I haven't been able to find a good explanation of it online. They are either like " You need 14 degrees in theoretical mathematics and two English degrees to understand this." Or parrot how fast this funtion grows. From what I can tell, the more trees you exhaust increases the number of more complicated trees with more options to hit in the future. If you want a function which is easier to prove but has the same sorta jig, the Google able term here is "Hydra Game"
But there is a roughly equivalent function on the Fast Growing Hierarchy: f_θ((Ω^ω)xω,0)(n) which is much faster growing than Graham's Function. Approximated by representing the TREEs as transfinite ordinals and comparisons between it and the weaker function tree(n) [no uppercase].
Yeah I know this topic has been covered a lot on UA-cam, but I chose this project as my first one, that way I could look at other people’s content for reference/inspiration. But lemme know what you wanna see next!
@@legendgames128 Rayo's number value depends upon a choice of a model of ZFC (or some other set theory). There is not a canonical choice (in fact we don't even know if ZFC is consistent so there could be no model). If there are models of ZFC, you can create ones where Rayo's number is arbitrarily large.
The fast-growing hierarchy is the "standard" function used for googology. Generally, the (capital) W-Y Sequence is considered to be the strongest function currently avaialble.
Graham's number is g64, not g63. I'd also like to know how the man himself pronounced his surname because I've heard it both ways now. (Gray-am and Gram). A couple of interviews use the two-syllable version, but maybe he was being polite. He was American, and "Gram" is more common over there, at least for a certain food brand.
Nice video! I wonder how you managed to animate the text and numbers in such a way. I want to make a similar style video on my own, so that's why I ask. Thanks!
@@Himechinachae BB(N) itself is not computable for arbitrary N, as it measures the maximum output size of a halting Turing machine with N states; being able to compute BB would be related to solving the halting problem, which is known to be unsolvable. On the other hand, “puny” numbers such as G64 are computable by a Turing machine with surprisingly few states.
@@paologat If you use a theoretical "oracle" you can get the values. Or if you use a theoretical "infinite memory, flops, etc" computer, then you could also solve it. In which case, there could be a stronger fast growing function equivalent to the largest finite number that can be printed by the theoretical computer. Since it can simulate busy beavers, this function should grow much faster than it.
Where'd you get that from? TREE(3) is much, much larger. There are no known bounds, but it is far larger than some 2^^1000, or 2^^^^^1000. It's not even close. This number is also overrrated for its size despite there being much larger ones.
This is legit on par with a 3blue1brown video and it is CRIMINAL that this channel doesn't have more views/likes/subs. I'm sure this video took some time to make. You did an amazing job here, keep this up and you'll be getting more support in no time.
Thanks man, I’ve been making videos for a while, and this class project came up. So I thought it would be a perfect chance to learn manim. And yeah it took some time, but I gotta work on my time management skills…
@@PizzaManJoe We all do man. Keep at it and you'll get better. Doesn't matter how many steps back you take, as long as you move forward each day you'll do great.
Please don't... I'm not saying the video isn't good but don't say it's on par with 3B1B when all it is is a rehash of some Numberphile videos in a different format.
18 Edit - wait no. 218 Edit 2 - wow I was way off, you know about a lot more numbers than me. Amazing video btw, hopefully you'll get enough likes for a part II but I'm not sure how many that is I'm still working on what comes after 218 Edit 3 - breakthrough! 219. I'm catching up
Crazy cool content. If you’re looking for concepts or topics, try covering something more cosmic! Like string theory or something like that. Topology is cool too!
Okay but... at what point do you cross over from Countably Finite, to Countably _Infinite?_ What point does math itself say "you're done counting, it's sideways eight time baybee"...
This is the first time i've ever heard of a low bound for TREE(3), and it looks like that's an extremely weak lower bound, like how the lower bound for the grahams number theory thingy is like 14 or something. as far as i've heard TREE(3) most likely outgrows Conway chained arrow notation, as far as mathematicians know, and grahams number which is g64 not g63, can be represented as fitting somewhere between 3→3→64→2 and 3→3→65→2
i dont really understand what is the number of possible combinations of trees in TREE(3). If grahams number is G64, then this is G (187196 amount of Arrows of 3s)?
I heard somewhere that the biggest number we’ve come up with is called “big foot.” I don’t understand it at all, but it seems like there’s a “foot language” with its own rules, and foot(n) is the largest number which can be created doing certain things with that language… and big foot is that repeated a whole lot
This is a comment about LaTeX If you want to continue making videos with manim which has the classic "maths" font, then I want you to research some conventions about when to use which symbols. For example, f(x) is usually written with an italic f and an italic x, which is the default, why? Because of the general rule of thumb "variables/changing things are in italics" This rule also makes standard text non-italic That's why I felt a bit odd when I saw you write 100 to the googol but with googol in italics. This rule applies most commonly to stuff like the sine and cosine and log functions, so sin(x) is written with an upright "sin" and an italic "x" But there's some debate when it comes to other common constants, like "e"; Euler's number, "π", "i"; the imaginary constant which is the(?) solution of i² = -1 Some write them upright because they are defined just like sin and cos and log But some write them in italics because there's many 'e's and 'π's and 'i's in formulas This comment is over.
If you say "in the limit" about some expression, all that matters is what happens with the expression beyond even the most unfathomable of the unfathomable numbers imaginable. Anything before then is irrelevant. So the first time you are asked to compute a limit as a student in a high school math class, you are already working with numbers way past any numbers that appear in this video.
The question was can I think of a bigger number, and that would be a good one Well, it is simply bigger than the Tree one, but also if we speak of fast growing functions, Tree() is popular, but pretty unremarkable, there are computable functions both above and beyond it. BB is more interesting in that regard, as it's the best upper bound for computable functions (as far as I know, maybe I'm mistaken)@@alt_meta3077
I made a lil game days ago in which i just made fun to remove bore in my mind. I just made a number idk how big, but the notation was this. Let ((2^64-1) x 4 +2) be Z And 2^64-1 be a Z^a^a^a.....^a In which the Z is raised to a power tower of a's that repeat a times. Im kinda lazy to tell the whole game's mechanics and the reason why Z is that crap, but ill tell it sooner when im energized xd
Start with Rayo number, Rayo(10^100). a = Rayo(Rayo(10^100). b is Rayo(Rayo(Rayo(a))). c is Rayo(Rayo(Rayo(Rayo(b))))...z...A...Z...alpha...omega... Alpha...Omega. Repeat this process Omega more times.
"omega" is a set cardinality aleph0. It does not make sense to iterate something an infinite amount of times. Furthermore, RAYO(n) is ill-defined due to its dependance on which set of axioms are used to contruct your chosen set theory. You could use ZFC, KP, etc..
Both BEAF and Bird's notation are ill-defined past f_{e0} (n) with respect to the fast growing hierarchy, and where e0 is an ordinal with the fixed point a->w^a. (w is the set of cardinality aleph0).
@@Galinaceo0isn't it just the biggest number that you can write using a google of symbols that are effective at writing down very big numbers in a small amount of their characters?
the reading of the stacking exponents is not quite right, 10^(10^10) and (10^10)^10 are totally different numbers and the former is read "ten to the ten to the tenth power power" and the latter "ten to the tenth power to the tenth power"
I am commenting #BringBackDislikes on every unique UA-cam video that I watch for the rest of 2024, regardless of if I actually dislike the video or not. This is video 3118.
How do you define absolute infinity? Also from the context i think its clear they are talking about natural numbers. Whatever you mean by absolute infinity is not a natural number.
If you know fast growing hiarchy, f(w^2) is the graham sequence. For TREE (3) however, none of the ordinals are powerful enough with function f. We need theta as a function, with omega as a value power by w. That is TREE(3). So its way bigger
All of that is wrong. f_{w^2}(64) is *far* larger than G64. It's not even close. The upper bound for G64 is f_{w+1}(64). Furthermore, TREE(n) has no official bounds. Though, it's supposedly not meant to exceed a growth rate of f_{LVO}(some reasonably large n). Finally, you did not even attempt to clarify "Theta", which I believe you are referencing feferman's theta function, and th(w^w) doesn't make any sense without a diagonalising function.
holy shit this is a school project????? dude this is actually phenomenal. like ive seen school projects in the past that are really good (like the one zaza made about eateot) but this is probably the best one ive seen
Thanks, I spent way too much time working on this, and I may or may not have pulled an all nigher the night before it was due…
@@PizzaManJoe you pulled an allnighter didnt you. lol.
@@tristantheoofer2 HI TRISTAN
@@eastonrocket3273 no way hello
all nighter*
Cackling at the last citation in the bibliography being the dancing skeletons. '😭'
The tree function only sounds like "just trust me bro"
yeah it kinda does, but its proven to be finite and ridiculously large so it works
Yeah. Same here. I haven't been able to find a good explanation of it online. They are either like " You need 14 degrees in theoretical mathematics and two English degrees to understand this." Or parrot how fast this funtion grows.
From what I can tell, the more trees you exhaust increases the number of more complicated trees with more options to hit in the future. If you want a function which is easier to prove but has the same sorta jig, the Google able term here is "Hydra Game"
But there is a roughly equivalent function on the Fast Growing Hierarchy: f_θ((Ω^ω)xω,0)(n) which is much faster growing than Graham's Function.
Approximated by representing the TREEs as transfinite ordinals and comparisons between it and the weaker function tree(n) [no uppercase].
It may just be a rehashed numberphile video, but i really like where this channel is going! Keep it up! :D
Yeah I know this topic has been covered a lot on UA-cam, but I chose this project as my first one, that way I could look at other people’s content for reference/inspiration. But lemme know what you wanna see next!
I was going to give you TREE(3) likes but my mouse melted.
On day one my Tree grew 1 meter. On day two he grew 3 meters. How long is he on day 3?
Great video! Keep making the interesting content.
Let me know what I should make next!
"Am I a joke to you?"
-- Rayo's Number
Yes, I say Rayo(Rayo(Rayo(Rayo(Rayo(10^100)))))
Rayo's number is not well defined.
@@appybane8481Rayo(Rayo(…Rayo(Rayo(10^100)))) iterated Rayo(10^100) times
@@Galinaceo0 ???
@@legendgames128 Rayo's number value depends upon a choice of a model of ZFC (or some other set theory). There is not a canonical choice (in fact we don't even know if ZFC is consistent so there could be no model). If there are models of ZFC, you can create ones where Rayo's number is arbitrarily large.
Do more googology!
I want to hear about the meanings of stuff much more powerful than the TREE function
SSCG(N)
The UA-cam channel "Orbital Nebula" has pretty good videos
The fast-growing hierarchy is the "standard" function used for googology.
Generally, the (capital) W-Y Sequence is considered to be the strongest function currently avaialble.
@@alt_meta3077 What are those functions?
This video will hit 1 million views for sure
Graham's number is g64, not g63. I'd also like to know how the man himself pronounced his surname because I've heard it both ways now. (Gray-am and Gram). A couple of interviews use the two-syllable version, but maybe he was being polite. He was American, and "Gram" is more common over there, at least for a certain food brand.
He started at G0 instead of G1, so it evens out.
@@neoieo5832👈🤓
@@neoieo5832 G0 is by definition equal to 4.
@@alt_meta3077 If you define G0 as 3^^^^3, then G63 would be Graham's number.
@@neoieo5832 Yes it would, except g0 is defined as 4.
Nice video! I wonder how you managed to animate the text and numbers in such a way. I want to make a similar style video on my own, so that's why I ask. Thanks!
It’s a python library called manim. I recommend checking out the documentation at www.manim.community
10/10 video, this is the quality I expect form channels with millions of subscribers, not hundreds.
I thought of busy bever numbers when you asked for a large number, aren’t they super large? Don’t recall
Busy Beaver numbers initially grow more slowly, but they quickly surpass any other (computable) sequence.
@@paologat is it computable?
@@Himechinachae BB(N) itself is not computable for arbitrary N, as it measures the maximum output size of a halting Turing machine with N states; being able to compute BB would be related to solving the halting problem, which is known to be unsolvable.
On the other hand, “puny” numbers such as G64 are computable by a Turing machine with surprisingly few states.
BB(10^100) >>>>> TREE(10^100)
@@paologat If you use a theoretical "oracle" you can get the values.
Or if you use a theoretical "infinite memory, flops, etc" computer, then you could also solve it.
In which case, there could be a stronger fast growing function equivalent to the largest finite number that can be printed by the theoretical computer. Since it can simulate busy beavers, this function should grow much faster than it.
this is insane, the edittings very nice, good work brotha!
Grahams number is G64, not G63.
only if you start at g1, not at g0 like he started. Definition shown in the video is correct, just offset.
@@melwin2251 g_0 is defined as 4. There is no offset, that is the standard.
This conetent is what makes me happy. Pls keep making videos like that!
Thx lemme know what other topics you would like me to cover
Tree(3) needs 2^^1000 mathematical symbols to symbolise it
watch me do it in 7:
Tree(3)
There, done
How do people even prove that like what, I have so many questions about big numbers
@@cesarcampuzanomartinez8182 fr
This is wrong. That is the number of symbols to prove it is finite.
Where'd you get that from?
TREE(3) is much, much larger. There are no known bounds, but it is far larger than some 2^^1000, or 2^^^^^1000. It's not even close.
This number is also overrrated for its size despite there being much larger ones.
hidden gem channel.
Greeting from Venezuela!
NO WAY THIS IS A SCHOOL PROJECT??? Dude you have huge potential!! Pursue youtube and see how it goes.
Do a video on you memorizing pi to 314 digits in 5 hours and if you fail you get pizza but with… **blegh** olives…
This is legit on par with a 3blue1brown video and it is CRIMINAL that this channel doesn't have more views/likes/subs.
I'm sure this video took some time to make. You did an amazing job here, keep this up and you'll be getting more support in no time.
Thanks man, I’ve been making videos for a while, and this class project came up. So I thought it would be a perfect chance to learn manim. And yeah it took some time, but I gotta work on my time management skills…
@@PizzaManJoe We all do man. Keep at it and you'll get better. Doesn't matter how many steps back you take, as long as you move forward each day you'll do great.
@@PizzaManJoe fr
Please don't... I'm not saying the video isn't good but don't say it's on par with 3B1B when all it is is a rehash of some Numberphile videos in a different format.
@@Dalroc Don't what
As a fellow high schooler, i can agree that your video was very greatly made. Keep up the good work mate.
Love this video! Keep it up 👍👍
This video is underrated , GJ man hope you make it one day
Imagine if this video cut off at 0:04.
this is, in fact, very nice.
Guys we gotta get to TREE(3) likes!!!
good video. feels calming
your current sub count is 225, remind me in a year what it is cause its gonna skyrocket.
2:05 i was not prepared for that 👂
WHAT only 342 subs?!?!
18
Edit - wait no. 218
Edit 2 - wow I was way off, you know about a lot more numbers than me. Amazing video btw, hopefully you'll get enough likes for a part II but I'm not sure how many that is I'm still working on what comes after 218
Edit 3 - breakthrough! 219. I'm catching up
ONLY 175 SUBSCRIBERS?!?!?!
hello?? a SCHOOL PROJECT??? literally tell me you got extra credit on this or so god help me because holy shittt this is incredible-
I did get 100%, unfortunately no extra credit
@@PizzaManJoe good for you!!
SCG(N) is crazier than TREE(N), same applies for SSCG(N)
But the craziest is Rayo's Number
Crazy cool content. If you’re looking for concepts or topics, try covering something more cosmic! Like string theory or something like that. Topology is cool too!
What software did you use for this video?
So nice
Largest number is Large Number Garden Number defined by pibot
Okay but... at what point do you cross over from Countably Finite, to Countably _Infinite?_ What point does math itself say "you're done counting, it's sideways eight time baybee"...
Well, I'll add 1 with your number. It's bigger.
Gotta love those salad numbers!
You tell me first, I bet ill come up with a bigger one
i thought it was g64
This is the first time i've ever heard of a low bound for TREE(3), and it looks like that's an extremely weak lower bound, like how the lower bound for the grahams number theory thingy is like 14 or something. as far as i've heard TREE(3) most likely outgrows Conway chained arrow notation, as far as mathematicians know, and grahams number which is g64 not g63, can be represented as fitting somewhere between 3→3→64→2 and 3→3→65→2
G64 (Graham's number) < G65 < G66 ... < GG64 < GGGG...GGGG64 with G64 G's < Godgahlah < Gridgahlah < Kubikahlah < Terossol < Godgathor < Godtothol < Godtertol < Godtopol < Godhathor < Godheptol
i dont really understand what is the number of possible combinations of trees in TREE(3). If grahams number is G64, then this is G (187196 amount of Arrows of 3s)?
TREE(TREE(TREE(TREE(…(3)
With “TREE” being used TREE(3) times
I heard somewhere that the biggest number we’ve come up with is called “big foot.” I don’t understand it at all, but it seems like there’s a “foot language” with its own rules, and foot(n) is the largest number which can be created doing certain things with that language… and big foot is that repeated a whole lot
Big foot was proven to be ill defined meaning it has no valid value
BIG FOOT was proven to be ill-defined over 5 years ago
lol skeletons
Yeah lol that was a 5:30AM addition. I pulled an all nighter to get this done before it was due…
TREE^TREE(3) (3)
REPEATED TREES
wait what 250 views
Now add. "+1"
its simple let someone else think and just +1
😗
TREE(TREE(3))
This is a comment about LaTeX
If you want to continue making videos with manim which has the classic "maths" font, then I want you to research some conventions about when to use which symbols.
For example, f(x) is usually written with an italic f and an italic x, which is the default, why? Because of the general rule of thumb "variables/changing things are in italics"
This rule also makes standard text non-italic
That's why I felt a bit odd when I saw you write 100 to the googol but with googol in italics.
This rule applies most commonly to stuff like the sine and cosine and log functions, so sin(x) is written with an upright "sin" and an italic "x"
But there's some debate when it comes to other common constants, like "e"; Euler's number, "π", "i"; the imaginary constant which is the(?) solution of i² = -1
Some write them upright because they are defined just like sin and cos and log
But some write them in italics because there's many 'e's and 'π's and 'i's in formulas
This comment is over.
how in the world do you come up with "graums number"?
If you say "in the limit" about some expression, all that matters is what happens with the expression beyond even the most unfathomable of the unfathomable numbers imaginable. Anything before then is irrelevant. So the first time you are asked to compute a limit as a student in a high school math class, you are already working with numbers way past any numbers that appear in this video.
Graham's number is g64, not g63
Finite xeisovarithas solos
my own notation of addition: a[1]b
n level notation (my own): a[n]b
packed a[b[n]a]b: a[1][n]b
packed a[b[n]a]b (different notation): a[1,n]b
packed a[b[a[...[n]...×k]a]b: a[k,n]b
Notation(k)=notation k
a[b,a[b,a[b,a[...×k]]]]b (packed notation): a[k,[↓]]b
BEAF ahh ripoff that's ill-defined past e0.
Infinity minus 1
Little Biggedon
Is not well-defined and therefore is not a number
Have you heard of beavers? :)
Uncomputable and pretty irrelevant.
The question was can I think of a bigger number, and that would be a good one
Well, it is simply bigger than the Tree one, but also if we speak of fast growing functions, Tree() is popular, but pretty unremarkable, there are computable functions both above and beyond it. BB is more interesting in that regard, as it's the best upper bound for computable functions (as far as I know, maybe I'm mistaken)@@alt_meta3077
I made a lil game days ago in which i just made fun to remove bore in my mind. I just made a number idk how big, but the notation was this.
Let ((2^64-1) x 4 +2) be Z
And 2^64-1 be a
Z^a^a^a.....^a
In which the Z is raised to a power tower of a's that repeat a times.
Im kinda lazy to tell the whole game's mechanics and the reason why Z is that crap, but ill tell it sooner when im energized xd
Start with Rayo number, Rayo(10^100). a = Rayo(Rayo(10^100). b is Rayo(Rayo(Rayo(a))). c is Rayo(Rayo(Rayo(Rayo(b))))...z...A...Z...alpha...omega... Alpha...Omega. Repeat this process Omega more times.
"omega" is a set cardinality aleph0. It does not make sense to iterate something an infinite amount of times. Furthermore, RAYO(n) is ill-defined due to its dependance on which set of axioms are used to contruct your chosen set theory. You could use ZFC, KP, etc..
BRAZIL MENTIONED!!!!!!!!
(Bizzy Beaver)what about BB(BB(BB(BB(BB(BB(BB(BB(52!))))))))
or ryos number or ryo(BB(BB(BB(BB(BB(BB(BB(BB(52!)))))))))
BB is uncomputable, Rayo is ill-defined.
Also why 52! of all numbers?
3:46 *pentation, not "pentration"
simplify TREE(3)/TREE(4) 💀
approximately 0
I'm sorry... It looks like a great video, but you lost me at 0:13
0:42
Me: {10,4[1/_(1/_(1/_(...)2)2)2]2} repeating {10,1000[2]2} times
Replying to you: TRITAR
Both BEAF and Bird's notation are ill-defined past f_{e0} (n) with respect to the fast growing hierarchy, and where e0 is an ordinal with the fixed point a->w^a. (w is the set of cardinality aleph0).
You forgot a bunch of other numbers that are bigger. Like Rayo's number, SSCG(3), omega, epsilon(0) and more
Yeah I definitely wanted to get into those numbers, but sadly the time limit and due date for the project did not permit
TREE(3) likes for part 2 tho
@@PizzaManJoe yippie part 2 ;D
Rayo's number is not well defined.
@@Galinaceo0isn't it just the biggest number that you can write using a google of symbols that are effective at writing down very big numbers in a small amount of their characters?
the reading of the stacking exponents is not quite right, 10^(10^10) and (10^10)^10 are totally different numbers and the former is read "ten to the ten to the tenth power power" and the latter "ten to the tenth power to the tenth power"
nobody reads like this.
brackets would be specified if necessary.
As soon I hear a chair creeping I knew it was a rookie creator then I saw your sub count. You’re so underrated! Keep making these content
I was recording while my friends were hanging out behind me, I tried to keep them quit but I guess that one slipped through lol
I am commenting #BringBackDislikes on every unique UA-cam video that I watch for the rest of 2024, regardless of if I actually dislike the video or not. This is video 3118.
😮
i said BIG FOOT
BIG FOOT > FOOT > Rayos Number
both wrong and all of those numbers are ill-defined, therefore don't exist.
4:39 why did you pronounce graham so badly
Well I could say "absolute infinity", infinity is not a number but is a kind of number
I hate to do the "uhm actually", but uhm actually infinity is a concept, not a number or kind of number, as it meaning is similar to "unending"
Infinity is a relative term it can be used when one quantity is unfathomably larger (or smaller) than the other.
@@frendlyleaf6187 its true that the word infinity is used in that context in day to day speech, but it is technically wrong
How do you define absolute infinity? Also from the context i think its clear they are talking about natural numbers. Whatever you mean by absolute infinity is not a natural number.
There's technically no absolute infinity.
If you know fast growing hiarchy, f(w^2) is the graham sequence. For TREE (3) however, none of the ordinals are powerful enough with function f. We need theta as a function, with omega as a value power by w. That is TREE(3). So its way bigger
Most of the stuff you said is wrong and nonsensical
All of that is wrong.
f_{w^2}(64) is *far* larger than G64. It's not even close. The upper bound for G64 is f_{w+1}(64). Furthermore, TREE(n) has no official bounds. Though, it's supposedly not meant to exceed a growth rate of f_{LVO}(some reasonably large n). Finally, you did not even attempt to clarify "Theta", which I believe you are referencing feferman's theta function, and th(w^w) doesn't make any sense without a diagonalising function.
¹⁰10 😎
bro straight copied an already made video
which video is that, exactly?
@@melwin2251 It's mostly rehashed information from Numberphile and the likes.
TREE and Graham's sequence are relatively useless in googology.