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
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
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.
While this is a good simplification, proof theory (which is essentially what "looking for the biggest number" eventually (de)volves into) is actually much deeper. A lot of times, new machinery needs to be developed before a new proof system can be pushed to its limits, e.g. types added to the λ-calculus, making the resulting system much more powerful, proofs much more expressive, (and "the maximum number or proofs in the system," which is often times the "big number" you're looking for, much bigger) but also often throwing a wrench into things (type resolution is not recursively-enumerable, for example). Big-O notation is just straight up asymptotic behavior, making it much more boring by comparison :)
It really is about the degree of operation. Different degrees of algorithmic operation grow at such different rates that too large of a gap between those operations defeats the whole purpose of lesser ones.
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
I mean. I don't know about "red herring". Yes, the relevant fact about BB that makes it unuseful for this challenge is that it's an uncomputable function, but it's an interesting observation, and I somehow don't think this video is concerned with practical significance when the final result is a compressed lambda calculus representation of a function that iterates over every program in the strongly normalizing calculus of constructions with length less than that function's input. I didn't know that the value of BB(5) actually got proved in just this year, last time I saw references to the results for that value they were only speculated to be optimal. To me that's interesting information.
I think the confusion about BB arises from the fact that your stated objective is "the largest number for which a generating algorithm fits in a SMS", but what you are presenting is actually "the largest number for which I was able to find a generating algorithm that fits in a SMS"
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.
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
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
To clarify, there is no general algorithm that can generate BB(n) for a given n, regardless of computation time, even infinite. If you want to treat it like a computable function, you need to use something called an "Oracle Machine" which can sweep the halting problem under the rug. And as far as proofs, eventually there will reach an n such that BB(n) is not provable in ZFC, or in any specific proof system you choose there will eventually be a value of n where it can no longer be proved. So the concept may be well-defined, but the outputs are debatable.
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
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: ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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
Hey @CodeParade, i dont know if im supposed to ask, but can you make a lil game or simulation where you can throw numbers fractions and other mathematical stuff into each other? and every time you do, a sound effect and the sum, product, quotient, difference, etc pops up and the numbers you tossed at each other disappear.
Hey there!, Talk about trio sequence system! Or TSS which grows from multi exponent to passing fast growing ordinal Its sentence is TSS(n) It already passes bukholz in 700! We could spam it and make layers with 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.
The monster group cardinal is probably the best candidate. Not that it is the biggest natural, but because any other number can be beaten by another growth category.
Wow! Ph.D. in math and I couldn't follow it. By the way, there is an incredible amount of information already coded in the languages and symbols, so in a sense you are using much more information than in a text message.
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.
Werr running out of words to describe these ever growing number sequences. There's no practicality to such large numbers so it's really just a fun mental exercise.
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)
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.
Your question as stated is technically not well-defined. First, to have a correspondence between a string of digits and a video, you need some system of encoding. If you don't specify such a system as part of your question, any number can form the video you suggested: we can just define a function Decode(n) that's hardcoded to simply return your video for any input n. Second problem is that "a video of me taking my first steps as a baby" is pretty vague. If I take a video of you, set the resolution to 1x1, set the length to 1 frame, and make it black and white, the whole video is just 1 bit of information. So you'll need to be a little more precise. Third problem is that Graham's number is finite and not random. So, strictly speaking, the true probability of a specific video appearing is either 1 or 0. But you could ask the probability that, if we were to generate a random video of a certain size, that video appears somewhere in G. Which should be equivalent to what you intended in your question. If you were pick any standard video codec and have a specific video file to search for, the probability should be extremely, extremely close to 1, assuming that the digits of a random substring of G are themselves sufficiently random (which I think is the case but I'm not sure). Let's say your video is 10MB in size: that's 8*10^7 bits. If we were to have a random string of that many bits, the probability of it being your exact video are the same as flipping 8^10^7 coins in a row and getting heads each time, which has a probability of 1/(2^(8^10^7)). This is a pretty big number by any normal standard, but the fact that we can easily write it with regular exponentiation shows that it's nothing compared to our friend big G so it would be almost certain to occur. To speak even more generally, if you come up with a string X to appear in G, its probability will almost certainly fall into one of four categories: 1, extremely close to 1, 0, and extremely close to zero. G has so many digits that the chance of your chosen X being within any reasonable number of orders of magnitude would be pretty much unfathomable.
I'll assume your question is similar to the original infinite monkey question. Off the top of my head--if a number is normal and infinite then the probability of finding any finite string, regardless of length, is equal to 1. If the number is not infinite and not normal, then the probability of finding a given finite string of sufficient length tends toward zero.
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.
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
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.
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".
At that point, the number of symbols required to express the number would be totally immaterial, because (say) Loader's number/233 is pretty much equal to Loader's number. Really, this applies to any number once you get above the scale of 10^10^n.
@@rtg_onefourtwoeightfiveseven Seems like a simple ration is not going to cut it, but relating the two still sound interesting. We would need some kind of byte efficiency metric.
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
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.
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
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...
It has become increasingly clear why you were able to pull off developing 4 dimensional games
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!
While this is a good simplification, proof theory (which is essentially what "looking for the biggest number" eventually (de)volves into) is actually much deeper. A lot of times, new machinery needs to be developed before a new proof system can be pushed to its limits, e.g. types added to the λ-calculus, making the resulting system much more powerful, proofs much more expressive, (and "the maximum number or proofs in the system," which is often times the "big number" you're looking for, much bigger) but also often throwing a wrench into things (type resolution is not recursively-enumerable, for example). Big-O notation is just straight up asymptotic behavior, making it much more boring by comparison :)
It really is about the degree of operation. Different degrees of algorithmic operation grow at such different rates that too large of a gap between those operations defeats the whole purpose of lesser ones.
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.
I failed maths in high school, am studying linguistics, where I don’t need any maths and yet I find this super fascinating
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.
I mean. I don't know about "red herring". Yes, the relevant fact about BB that makes it unuseful for this challenge is that it's an uncomputable function, but it's an interesting observation, and I somehow don't think this video is concerned with practical significance when the final result is a compressed lambda calculus representation of a function that iterates over every program in the strongly normalizing calculus of constructions with length less than that function's input. I didn't know that the value of BB(5) actually got proved in just this year, last time I saw references to the results for that value they were only speculated to be optimal. To me that's interesting information.
I think the confusion about BB arises from the fact that your stated objective is "the largest number for which a generating algorithm fits in a SMS", but what you are presenting is actually "the largest number for which I was able to find a generating algorithm that fits in a SMS"
its like im watching a really dumb powerscaling video.
also always remember... all of these numbers are closer to 0 than to ∞
getting angry stares after saying that some person has a power level of Loader's Number
Damn, every single time I am researching something on the cusp of new Computer Science, John Tromp is always there
Matt Turk’s long lost broþer
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)
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)
Yeah, I used to, and still play the games of Patcail
Noobs, i play AM (totally without timewalls)
6:46 My mind passed that point a while ago
I didn't understand anything from the first video
Can't wait till we see Code Parade's new "orders of orders of magnitude" game haha.
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
What I hear: Loader's number
My mind: Overloader's number
This is my favorite type of videos. please keep it coming!!
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
@@shophaune2298what an arbitrary choice to make
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
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.
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.
Glad that my comment inquiry regarding BMS in the first video was considered. Great vid
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
To clarify, there is no general algorithm that can generate BB(n) for a given n, regardless of computation time, even infinite. If you want to treat it like a computable function, you need to use something called an "Oracle Machine" which can sweep the halting problem under the rug.
And as far as proofs, eventually there will reach an n such that BB(n) is not provable in ZFC, or in any specific proof system you choose there will eventually be a value of n where it can no longer be proved. So the concept may be well-defined, but the outputs are debatable.
everyone is gangsta until the notation for representing ordinals changes
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.
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
ad ends at 3:44
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: ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.
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
Just wanna say thanks for your videos!
thank you Discrete Mathematics for giving me the tools to understand this a lil' bit.
Fun fact the BB for busy beaver actually stands for busy Beaver which is pretty cool and also (TREE(∑(⁹9!↑↑↑↑⁹9!↑↑↑↑⁹9!))) is a pretty big number.
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
Hey @CodeParade, i dont know if im supposed to ask, but can you make a lil game or simulation where you can throw numbers fractions and other mathematical stuff into each other? and every time you do, a sound effect and the sum, product, quotient, difference, etc pops up and the numbers you tossed at each other disappear.
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
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
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
Wake up babe new code parade vid just dropped
Yay
My brain is too smooth for this. I need to be immortal to understand this but still was an interesting watch
>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!
Hey there!, Talk about trio sequence system! Or TSS which grows from multi exponent to passing fast growing ordinal
Its sentence is TSS(n)
It already passes bukholz in 700! We could spam it and make layers with it
wait, PATCAIL!? the one who made that one incremental game i played!? didn't expect to hear that name on here!
I fear the game that is going to come out of this series of videos.
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
Finally the sequel came out
Hell yeah. This day just got better
The monster group cardinal is probably the best candidate. Not that it is the biggest natural, but because any other number can be beaten by another growth category.
Wow! Ph.D. in math and I couldn't follow it. By the way, there is an incredible amount of information already coded in the languages and symbols, so in a sense you are using much more information than in a text message.
was waiting for this
Proving the output of a function is crazy
4D golf would go crazy on VR, would definitely recommend trying to port it
Would BMSw in base TREE 3 be larger than Loader's number?
Fun fact: Patcail made an incremental game about ordinals called Ordinal Markup
that sure is more likes than i ever got
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.
Werr running out of words to describe these ever growing number sequences. There's no practicality to such large numbers so it's really just a fun mental exercise.
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?
Oh hey Patcail. I know him. He made that funny ordinal game Ordinal Markup :D
Great! Like I said, a followup video was always possible!
Where is the Brilliant course that will help me understand the numbers in this video?
BMS mention LET'S GOOOOOOOOOOOOO
aint no way this is my motivation to study PTOs
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)
It'd be the same function, just with a bigger input.
If ψ₀ (Ω) ascends beaf notation, then it is part of a infinite growing notation (FGH)?
Can you make a video on the greatest cardinals higher than inaccessible?
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.
Your question as stated is technically not well-defined. First, to have a correspondence between a string of digits and a video, you need some system of encoding. If you don't specify such a system as part of your question, any number can form the video you suggested: we can just define a function Decode(n) that's hardcoded to simply return your video for any input n.
Second problem is that "a video of me taking my first steps as a baby" is pretty vague. If I take a video of you, set the resolution to 1x1, set the length to 1 frame, and make it black and white, the whole video is just 1 bit of information. So you'll need to be a little more precise.
Third problem is that Graham's number is finite and not random. So, strictly speaking, the true probability of a specific video appearing is either 1 or 0. But you could ask the probability that, if we were to generate a random video of a certain size, that video appears somewhere in G. Which should be equivalent to what you intended in your question.
If you were pick any standard video codec and have a specific video file to search for, the probability should be extremely, extremely close to 1, assuming that the digits of a random substring of G are themselves sufficiently random (which I think is the case but I'm not sure). Let's say your video is 10MB in size: that's 8*10^7 bits. If we were to have a random string of that many bits, the probability of it being your exact video are the same as flipping 8^10^7 coins in a row and getting heads each time, which has a probability of 1/(2^(8^10^7)). This is a pretty big number by any normal standard, but the fact that we can easily write it with regular exponentiation shows that it's nothing compared to our friend big G so it would be almost certain to occur.
To speak even more generally, if you come up with a string X to appear in G, its probability will almost certainly fall into one of four categories: 1, extremely close to 1, 0, and extremely close to zero. G has so many digits that the chance of your chosen X being within any reasonable number of orders of magnitude would be pretty much unfathomable.
I'll assume your question is similar to the original infinite monkey question. Off the top of my head--if a number is normal and infinite then the probability of finding any finite string, regardless of length, is equal to 1. If the number is not infinite and not normal, then the probability of finding a given finite string of sufficient length tends toward zero.
what about the weakly compact cardinal? it is far larger than the buchholz ordinal or loaders number
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.
wait. patcail? like, the guy who mode ordinal markup?
yup.
me not comprehending anything and just accepting the "certified largest number" as what he says it is.
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
Now all we need is a large number-finding game 👍
Around 2:25 what are all those notations like 'omega growth' up to 'TFBO Growth'?
do the omegas and absolute infinitys
bro you only mention oblivion and utter oblivion once in the video 😭
Wow. Still not as big as my… uh, my uh… my lose streak in video games
💀
Loader's number times Loader's number.
There is still a ordinal that can fit in 1 character: Ω/Omega Capital
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.
Oh so that's what you call people attracted to CoC, googologists
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".
If it takes forever to compute even in the theoretical sense, then it's not computable.
You can just do BIG FOOT, Sam's Number, Utter Oblivion, Ultimate Oblivion, and if you REALLY want a big number, do phi omega of 0.
Well, JonTron joke is the only thing I understood from this video.
What video is the "Utter Oblivion" thumbnail from? I tried searching for it, but can't find it.
You could fit much more binary lambda calculus in a text using base32 or base64 encoding of the final program binary, just saying
Imagine the tree number getting multiplied by itself like tree (tree 3)
Ultimate oblivion is the biggest number I have found before infinity…
But why are there so many numbers bigger than infinity WHY!
Because is FICTIONAL GOOGOLOGY
oh damn thats me! cmon bro bigfoot is a awesome name tho
Large Garden number is bigger it even makes infinity seem small.
The number of atoms in omniverse 😊
Wow, loaders number is really big.
How about a ratio between the number of symbols to express the number and the number itself?
At that point, the number of symbols required to express the number would be totally immaterial, because (say) Loader's number/233 is pretty much equal to Loader's number. Really, this applies to any number once you get above the scale of 10^10^n.
@@rtg_onefourtwoeightfiveseven Seems like a simple ration is not going to cut it, but relating the two still sound interesting.
We would need some kind of byte efficiency metric.
remind me why am I listening to a guy talking about obscure math trying to fit a big number in an SMS?
And it's still closer to 0 than infinity
It’s infinitely closer to 0 than infinity
What are we doing finding the largest number? Just taking that and make a fraction out of it to make the "smallest" number?