I don't think I have ever watched a UA-cam video where I understood so little of it . The number of terms and concepts to look up recursively to understand these numbers in detail is almost as large as the numbers themselves
Never have I realized how difficult googology is to find your way around in, especially in deeper parts like this I mean, the jargon in this video is insane
If anyone is confused why busy beaver numbers don't work: It's basically the same as saying "the largest number that can fit in a text message is the largest number that can fit in a text message"
Almost correct but wrong explanation. The reason bb can be use is because you don't know. By that I mean it is uncomputable. Or you don't know what number it is. And it can also change
@@nodrance for what I know there are numbers that fall more in the philosophical area than in the Maths one. That concept of the “largest numbers that fits” sometimes feels more logical but for another science. That’s why Rayo (eho is a philosopher) created his own big number
Alright so from what I can gauge number classes aren't necessarily literal numbers with predefined digits. They're more comparable to Big O Notation where you simply identify what part dominates as n approaches infinity. For example, if you ever told a CompScientist "O(n^2 + 1) is greater than O(n^2)" you'd be laughed at because the rate at which O(n^2) grows makes that +1 so irrelevant there's no reason in specifying.* It's why the notation is rather simple to begin with. If you have a growth rate of a polynomial with a number of degrees up to 1000, degrees 0-999 are discarded. And even that is dwarved by any exponential function with a base larger than 1. The only difference is we've transcended shit like exponential, factorial, and O(n^n)--and that last one is already pushing it because any program with that bad of Big O is either so bad to never be even used, or pumped full of tiny optimizations that try to withstand the inevitable rampant growth for just long enough to get something useful. *To those who don't quite get what I mean, lets start simple. n^2 vs n^2+1 when n = 2 is 4 and 5. That +1 provides a 25% increase, which is pretty significant. However, n = 3 is 9 vs 10, which only ~11%. As n grows, that percentage increase shrinks to insignificance. So when it comes to Big O notation, we don't really give a shit about +1. This is true for any inequal growth. for example n^3 vs n^3 + n^2 are considered equivalent under this notation because when n = 2, you get 8 vs 12. Although that's a 50 percent increase, n = 3 gives 27 vs 36 which is only a 33% increase. When n = 10 that difference is only a 10% increase. Every time you double n, the percentage increase is half. n = 20 is +5%. n = 40 is +2.5%. n = 80 is +1.25%. et cetera. So you quite literally disregard everything that's not the leading value because it's basically a diminishing return.
Reading about Graham's Number and other large numbers in the past made me appreciate how you never get close to infinity, even if sometimes it can feel like a big number could just be equated to infinity. Climbing the ladder in defining incredibly large numbers while satisfying some constraints is still fun though.
Mentioning that the busy beaver numbers are difficult to compute because they are so large and that we will probably never know the value of BB(6) is a red herring. These numbers are all too large for anything anyway. The qualitatively different property that the busy beaver sequence has is that it is uncomputable and the rest doesn't matter.
the problem with the busy beavers is just that theyre not something with a function, theyre just a placeholder for the idea of a biggest possible number
Isn't the BB(n) function in this case similar to a hypothetical MLC(n) function that is "the biggest number that can be written in lambda calculus using n symbols"?
I mean, that becomes very philosophical very quickly. It's totally possible that it's impossible to prove exactly what value of BB(n) for some n. So then you're basically at a tree falling in the forest
this is now the thrid different ruleset i have heared about the hydra game, there goes my weekend trying different trees and writing code to solve them
No way! Patcail! That used-to-be huge bastard! I'm a mod in his ( now dead ) discord server, and those were some years, i'll tell ya. Also, haven't seen him in years, never expected to see him again
Hearing that your son is taking freaking Brilliant courses was quite the reality check for me, as in my mind he's always been the adorable toddler climbing the DIY rockwall. 😏
still no mention of unary I see. The true largest number that can fit in 140 characters (given the stipulation that it must be computable without outside information) is 140, expressed like this: ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Correct me if im wrong, but wouldn't one have to somehow analytically prove that a shit ton of Turing machines never halt to then compute the BB? Or have they developed some crazy new methods?
@@Pizhdak they've programmed deciders which looked for patterns in the behavior of 5-state turing machines, and ruled out any machines running for more than 47176870 steps as non-terminating
Now to find the most awkward numbers: define the most awkward number of n to be the least natural number that's not expressed as any lambda calculus expression of size n or lower (obviously awkward(n)an+b for some a and b since you have an exponential bound on the program count and you can just write down the number naively) kinda interesting to know what those are, but also they are presumably uncomputable
Rayo's Number is kinda cringey because of the arbitrary use of a googol as the parameter. I wonder if there's a more natural big number to use for this kind of construction.
Loader's Number is a kind of cheat because it was made as a submission for the competition to return the largest numner that a C program within 512 characters can return. The differences with Busy Beaver function is just that the program is actually written out and we're not sure if it's the theoretical winner.
The next step here would be to remove the arbitrary restrictions on text length, for we live in a finite observable universe. How large is the largest number using all atoms in the universe to represent it? How about all particles in the universe? All permutations of planck units?
in terms of computable numbers that's still going to be Loader's number, I believe. If you mean the largest possible number under those constraints, then we're looking at Rayo's number (which is uncomputable, it declares itself as the largest number less than a googol symbols - approximately the number of subatomic particles in the universe - without providing a means to calculate it)
Wouldn't it be more precise to talk about finding functions that scale faster than other functions. That would automatically satisfy the requirement of having a way to generate the number and only caring about number classes.
BB(n) is a uncomputable function, just not in your sense. BB(n) is a searching function, search a Turing machine that output a langest string of 1 that is terminated. The uncomputable sense is it gonna take forever to compute.
Yeah I feel there is a confusion here between uncomputable functions, and uncomputable numbers. While BB(n) is an uncomputable function, I'm pretty sure that BB(n), for a specific n, is not an uncomputable number.
Finding BB(n) is not limited by computational power, you can't just leave a computer running and get an answer. The problem is, you have programs running and you can't tell if the program will end with a massive number, or never end. For example, imagine your program iterates all numbers and returns the first number that doesn't reach the 1-2-4 loop of the Collatz conjecture. That might be a *really* large number, or it might run forever, but you won't know which unless you prove or disprove the Collatz conjecture first. Likewise, finding BB(n) involves finding proofs to tons of math problems like that, it can't be computed by just leaving a computer running. That's why it's called "uncomputable".
What I want to know is the likelihood of whether a number contains a known string. For example, what is the probability that Graham’s number contains a string of digits that form a video of me taking my first steps as a baby? I want to see numbers classified in this way.
I could write a function that would type out the symbols to make up Rayo's number, even if I couldn't compute it. Even that would probably take longer than the age of the universe to complete, but I could do it.
I don't understand why in rule 2 demonstration, when replacing the right branch with the entire tree, the left branch also gets replaced (and this doesn't seem to happen in subsequent steps?)
does the left branch in that first step actually count as the right branch because it started out as one at the beginning of the game? And so there are two "right branches"?
No, that doesn't seem right because the same thing happens in the next step and the left branch (which was right at the start) is left alone. I'm still confused.
If I would choose a Bigger num (doesn’t matter that there are bigger ones) I would choose something that needs Babel Library Possible Books arrangement (Borges Cited) ~ 1M x 10^10^1,000,000 Bytes in BLC. curious that BL is the initials for both Babel’s Library and Binary Lambda. From now it would sound uncomputable-ish but I would choose this New Number order.
I'd just go TREE(TREE(TREE(...(TREE(Graham's number)))...) where the amount of TREE is Graham's number and then raise it to the power of the same thing(tree tree tree... grahams number). IK that is very bland.
BMS make small numbers in my opinion so i made my own matrix system CMS. CMS can surpass BMS really a lot without even getting any close to its limit... And maybe it can even surpass loader's number... if you want an explanation of how CMS works... DO 1000 LIKES!
The googology community is up in arms for receiving a measly "huge" thanks
🤣
huge could semantically mean anything from 2 to loader's number lol
“A googological thanks to the googology community.”
@@zyansheepI don't know if anyone would connote 2 with being 'huge', but it's hard to say where the line should really be.
@@jblen what if it's a p-value
I don't think I have ever watched a UA-cam video where I understood so little of it . The number of terms and concepts to look up recursively to understand these numbers in detail is almost as large as the numbers themselves
I feel you, I started to doubt if I really am fluent in english watching this
I need 2 hours video of explainging what actually these are
but at least its finite and computable
@@megadeth116 orbital nebula's series exists.
New biggest number - the recursive number of steps required to understand the previous biggest number
Never have I realized how difficult googology is to find your way around in, especially in deeper parts like this
I mean, the jargon in this video is insane
If anyone is confused why busy beaver numbers don't work: It's basically the same as saying "the largest number that can fit in a text message is the largest number that can fit in a text message"
r/TechnicallyTheTruth
Almost correct but wrong explanation.
The reason bb can be use is because you don't know.
By that I mean it is uncomputable. Or you don't know what number it is.
And it can also change
It's not the same, what are you talking about? You can define busy beaver numbers, you just can't prove what they are except for very small inputs.
@@nodrance for what I know there are numbers that fall more in the philosophical area than in the Maths one. That concept of the “largest numbers that fits” sometimes feels more logical but for another science. That’s why Rayo (eho is a philosopher) created his own big number
No, they are not written with human languages but in math symboles, so this paradox does not exist.
You say huge thanks, but what class of huge are you talking about?
recursively: the smallest class of huge which is larger than the class of huge you thought it was, minus one
gap ordinal level
Damn, changed my mind: Gotta be at least 5
5+1
Checkmate atheists
@@WaffleAbuser lol
@@WaffleAbuserthats not a jumber, that's a summ, obviously. Nothing's larger than 5
that’s underestimation, it’s gotta be atleast 9
@@spaceguy20_12I’d say that it’s at least 11, I don’t know really.
Loader's number mentioned. I forgive part 1 now.
Man, this video is inspiring me to get back into googology
At the time part 1 was made, Loader hadn't be made to fit in a tweet yet...
Alright so from what I can gauge number classes aren't necessarily literal numbers with predefined digits. They're more comparable to Big O Notation where you simply identify what part dominates as n approaches infinity. For example, if you ever told a CompScientist "O(n^2 + 1) is greater than O(n^2)" you'd be laughed at because the rate at which O(n^2) grows makes that +1 so irrelevant there's no reason in specifying.* It's why the notation is rather simple to begin with. If you have a growth rate of a polynomial with a number of degrees up to 1000, degrees 0-999 are discarded. And even that is dwarved by any exponential function with a base larger than 1. The only difference is we've transcended shit like exponential, factorial, and O(n^n)--and that last one is already pushing it because any program with that bad of Big O is either so bad to never be even used, or pumped full of tiny optimizations that try to withstand the inevitable rampant growth for just long enough to get something useful.
*To those who don't quite get what I mean, lets start simple. n^2 vs n^2+1 when n = 2 is 4 and 5. That +1 provides a 25% increase, which is pretty significant. However, n = 3 is 9 vs 10, which only ~11%. As n grows, that percentage increase shrinks to insignificance. So when it comes to Big O notation, we don't really give a shit about +1. This is true for any inequal growth. for example n^3 vs n^3 + n^2 are considered equivalent under this notation because when n = 2, you get 8 vs 12. Although that's a 50 percent increase, n = 3 gives 27 vs 36 which is only a 33% increase. When n = 10 that difference is only a 10% increase. Every time you double n, the percentage increase is half. n = 20 is +5%. n = 40 is +2.5%. n = 80 is +1.25%. et cetera. So you quite literally disregard everything that's not the leading value because it's basically a diminishing return.
Yes, that's exactly right! Big O is the same concept in computer science.
I was thinking the same thing!
Reading about Graham's Number and other large numbers in the past made me appreciate how you never get close to infinity, even if sometimes it can feel like a big number could just be equated to infinity. Climbing the ladder in defining incredibly large numbers while satisfying some constraints is still fun though.
It has become increasingly clear why you were able to pull off developing 4 dimensional games
Damn, every single time I am researching something on the cusp of new Computer Science, John Tromp is always there
Mentioning that the busy beaver numbers are difficult to compute because they are so large and that we will probably never know the value of BB(6) is a red herring. These numbers are all too large for anything anyway. The qualitatively different property that the busy beaver sequence has is that it is uncomputable and the rest doesn't matter.
the problem with the busy beavers is just that theyre not something with a function, theyre just a placeholder for the idea of a biggest possible number
@@danger_1189 What? It's a well defined function from N to N.
5:58 PATCAIL! Wow, I only know so much about large number because I played their games, nice to see them come up here
Yeah, patcail's certainly a name
i literally watched this while waiting on an ordinal markup timewall lol (grinding singularity levels)
Noncomputable ≠ not well defined, BB(n) is just a function from ℕ→ℕ, it's just impossible to observe in finite time
Isn't the BB(n) function in this case similar to a hypothetical MLC(n) function that is "the biggest number that can be written in lambda calculus using n symbols"?
@@akeem2983 yes as untyped lambda calculus ≅ turing machines, however it's still a well defined function
I mean, that becomes very philosophical very quickly.
It's totally possible that it's impossible to prove exactly what value of BB(n) for some n.
So then you're basically at a tree falling in the forest
@@johngalmann9579 I mean, we can trivially prove the value does exist. It's a value hand-picked by God himself but still exists
I failed maths in high school, am studying linguistics, where I don’t need any maths and yet I find this super fascinating
Fun fact: Patcail made an incremental game about ordinals called Ordinal Markup
2:40 Oh… (a) that actually makes the challenge meaningful now, and (b) I wish more people mentioned this
Yea, i also just heard of it for the first time, although i had a guess it is so, because otherwise you could always say +1
I wonder what the strict definition of a class is though
this is now the thrid different ruleset i have heared about the hydra game, there goes my weekend trying different trees and writing code to solve them
6:46 My mind passed that point a while ago
BB(n) and some faster-growing functions can be defined using a program but it require solving the halting problem to be computed, which is impossible.
After I have studied Googology for a few months I could actualy follow your video and also it help me understand a lot of things in the end.
Glad that my comment inquiry regarding BMS in the first video was considered. Great vid
No way! Patcail! That used-to-be huge bastard! I'm a mod in his ( now dead ) discord server, and those were some years, i'll tell ya.
Also, haven't seen him in years, never expected to see him again
Can't wait till we see Code Parade's new "orders of orders of magnitude" game haha.
5:20 isn't stackoverflow, it's code golf! That's exactly what you're doing too! Code golf is such a niche but awesome game
Hearing that your son is taking freaking Brilliant courses was quite the reality check for me, as in my mind he's always been the adorable toddler climbing the DIY rockwall. 😏
ad ends at 3:44
still no mention of unary I see. The true largest number that can fit in 140 characters (given the stipulation that it must be computable without outside information) is 140, expressed like this: ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
A truly stunning result, can’t believe he never brought this up
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII*IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII is bigger, and that doesn't even fit in the max
And I'd say that by rule 3 if the Part 1 video, this is the most basic
@@cewla3348you must define * first
@@Syuvinya You must define | first.
1:41 IT WAS PROVEN???
Yeah it's weird there wasn't more of a fanfare
a couple weeks ago yeah
Yes it just was
Correct me if im wrong, but wouldn't one have to somehow analytically prove that a shit ton of Turing machines never halt to then compute the BB? Or have they developed some crazy new methods?
@@Pizhdak they've programmed deciders which looked for patterns in the behavior of 5-state turing machines, and ruled out any machines running for more than 47176870 steps as non-terminating
Now to find the most awkward numbers:
define the most awkward number of n to be the least natural number that's not expressed as any lambda calculus expression of size n or lower
(obviously awkward(n)an+b for some a and b since you have an exponential bound on the program count and you can just write down the number naively)
kinda interesting to know what those are, but also they are presumably uncomputable
Rayo's Number is kinda cringey because of the arbitrary use of a googol as the parameter. I wonder if there's a more natural big number to use for this kind of construction.
The only big number that'd seem "natural" would be ~10^82, the estimated number of subatomic particles in the universe.
@@shophaune2298 10^185 Planck Volume in the observable Universe
Loader's Number is a kind of cheat because it was made as a submission for the competition to return the largest numner that a C program within 512 characters can return. The differences with Busy Beaver function is just that the program is actually written out and we're not sure if it's the theoretical winner.
WAIT, THE 5 STATE BUSY BEAVER IS OUT NOW?!
Yep, the value shown in this video is the maximum number of steps (as opposed to the maximum number of 1s possible)
BMS mention LET'S GOOOOOOOOOOOOO
>says greedy clique sequences are not rigorously proven
>uses BMS as an example
There's a paper, the lower bound was proven recently.
@@CodeParadeReally? Cool!
everyone is gangsta until the notation for representing ordinals changes
The next step here would be to remove the arbitrary restrictions on text length, for we live in a finite observable universe. How large is the largest number using all atoms in the universe to represent it? How about all particles in the universe? All permutations of planck units?
in terms of computable numbers that's still going to be Loader's number, I believe. If you mean the largest possible number under those constraints, then we're looking at Rayo's number (which is uncomputable, it declares itself as the largest number less than a googol symbols - approximately the number of subatomic particles in the universe - without providing a means to calculate it)
My brain is too smooth for this. I need to be immortal to understand this but still was an interesting watch
Wouldn't it be more precise to talk about finding functions that scale faster than other functions. That would automatically satisfy the requirement of having a way to generate the number and only caring about number classes.
some functions have a minimum size to define them, which the size limit of a text helps constrain. so it's not entirely the same problem
Proving the output of a function is crazy
BB(n) is a uncomputable function, just not in your sense. BB(n) is a searching function, search a Turing machine that output a langest string of 1 that is terminated. The uncomputable sense is it gonna take forever to compute.
Yeah I feel there is a confusion here between uncomputable functions, and uncomputable numbers. While BB(n) is an uncomputable function, I'm pretty sure that BB(n), for a specific n, is not an uncomputable number.
Finding BB(n) is not limited by computational power, you can't just leave a computer running and get an answer. The problem is, you have programs running and you can't tell if the program will end with a massive number, or never end. For example, imagine your program iterates all numbers and returns the first number that doesn't reach the 1-2-4 loop of the Collatz conjecture. That might be a *really* large number, or it might run forever, but you won't know which unless you prove or disprove the Collatz conjecture first. Likewise, finding BB(n) involves finding proofs to tons of math problems like that, it can't be computed by just leaving a computer running. That's why it's called "uncomputable".
wait, PATCAIL!? the one who made that one incremental game i played!? didn't expect to hear that name on here!
Finally the sequel came out
Hell yeah. This day just got better
Uhhhhhh 4 that sounds pretty big
WAIT
I JUST THOUGHT OF 40
@@BetterCaulipowerSall-vq9yn what about 41 😎
@@chnhakk what the hell are you talking about
@@BetterCaulipowerSall-vq9ynIDK, I’ve been hearing pretty good things about this “45” number…
What about 54@@chnhakk
This is my favorite type of videos. please keep it coming!!
Ok loader's number + 1
I win every time...
that wouldnt fit in 140 characters
@@ataraxianAscendant lambda loader's number didn't fit in 140 characters
Great! Like I said, a followup video was always possible!
And it's still closer to 0 than infinity
from said Googology and Apeirology community. it's really cool to see our community get recognised by such a number of people :3
fr
@@vari6989 yess :3
What I want to know is the likelihood of whether a number contains a known string. For example, what is the probability that Graham’s number contains a string of digits that form a video of me taking my first steps as a baby?
I want to see numbers classified in this way.
I fear the game that is going to come out of this series of videos.
was waiting for this
That number plus 1
I could write a function that would type out the symbols to make up Rayo's number, even if I couldn't compute it. Even that would probably take longer than the age of the universe to complete, but I could do it.
There is still a ordinal that can fit in 1 character: Ω/Omega Capital
Now all we need is a large number-finding game 👍
Wake up babe new code parade vid just dropped
Yay
I don't understand why in rule 2 demonstration, when replacing the right branch with the entire tree, the left branch also gets replaced (and this doesn't seem to happen in subsequent steps?)
does the left branch in that first step actually count as the right branch because it started out as one at the beginning of the game? And so there are two "right branches"?
No, that doesn't seem right because the same thing happens in the next step and the left branch (which was right at the start) is left alone. I'm still confused.
What this is asking for: "The largest number that does NOT fit into a text message" does fit into a text message and we get another fancy paradox.
If I would choose a Bigger num (doesn’t matter that there are bigger ones) I would choose something that needs Babel Library Possible Books arrangement (Borges Cited) ~ 1M x 10^10^1,000,000 Bytes in BLC. curious that BL is the initials for both Babel’s Library and Binary Lambda. From now it would sound uncomputable-ish but I would choose this New Number order.
Can you make a video on the greatest cardinals higher than inaccessible?
The kid who just adds 1:
the kid who adds a layer of f_ε_0:
its like im watching a really dumb powerscaling video.
also always remember... all of these numbers are closer to 0 than to ∞
bro you only mention oblivion and utter oblivion once in the video 😭
How can an axiom system like ZF or ZFC even have a countable proof theoretic ordinal if they can proof the exisitence of uncountable ordinals?
8:22 that’s sus
BBλ(1) = Lambda Busy Beaver One
wait. patcail? like, the guy who mode ordinal markup?
yup.
BBλ(2) = Lambda Busy Beaver Two
is it googleplex?
Wow. Still not as big as my… uh, my uh… my lose streak in video games
💀
zzzz... for 140 characters
z = to zzz.. for 140 characters
you see where im going with this
Wow, loaders number is really big.
Im going with the notation that includes 420
Math never existed.
It's true. I'm math, and I'm not real.
aint no way this is my motivation to study PTOs
What video is the "Utter Oblivion" thumbnail from? I tried searching for it, but can't find it.
Am incredibly confused but i like the video
Damn, that's like. Atleast 12.
What are we doing finding the largest number? Just taking that and make a fraction out of it to make the "smallest" number?
yes, I understood like 3 words after 5:30
JSON parsing to compute Bucholz Ordinals. Ouchies
tree(3) is unknown, so you cannot know if those are bigger than it
Absolute Aperdinal watching
Not first
Congratulations! You were first. You won 5 internet credits.
Also not last
if you want a big numbers just write the numbers very big
I'd just go TREE(TREE(TREE(...(TREE(Graham's number)))...) where the amount of TREE is Graham's number and then raise it to the power of the same thing(tree tree tree... grahams number). IK that is very bland.
why not find the smallest number greater than zero that can fit in a text message
1/answer
did you on that mathis r.v have able to reach beyond INFINITY!
Enters Infinity
Busy Beaver utter oblivion💀
No idea what any of this is but I watched it anyway
Loader’s number + 1
Pisslorgulus is a cool number
alright, now what's the smallest > 0 number you can make ?
BMS make small numbers in my opinion so i made my own matrix system
CMS. CMS can surpass BMS really a lot without even getting any close to its limit... And maybe it can even surpass loader's number...
if you want an explanation of how CMS works... DO 1000 LIKES!
Unsigned(-1)
i love googology
Oh so that's what you call people attracted to CoC, googologists