Notes and corrections: I mispronounced the atom cesium at the very beginning of the video, pronouncing it 'Kasium' I said that Omega ^ Omega x Omega is the same as Omega^ Omega ^ Omega when that's actually very wrong. At 6:11 I used a coefficient with an ordinal when really ordinal multiplication is non-commutative so that could cause problems. There are several minor phrasing errors around that amounts of alephs and omegas when I'm saying how long to wait. I had the original idea for this video ages ago when watching a Vsauce about infinity and noticing that it went past many of the ordinals. (Go and watch that video if you haven't, by the way, it's quite a bit more comprehensive than this one.)
‘There’s this mountain of pure diamond. It takes an hour to climb it and an hour to go around it, and every hundred years a little bird comes and sharpens its beak on the diamond mountain. And when the entire mountain is chiseled away, the first second of eternity will have passed.’ You may think that’s a hell of a long time. Personally, I think that’s a hell of a bird. (From Doctor Who)
Ever since vsauce made how to count past infinity 8 years ago, I've wanted to see another video that goes into more detail about the numbers larger than the ones he described, as he jumped almost straight from epsilon to the innacessable cardinals. I've finally found one. This is probably my new favorite video to do with numbers in general.
It's rather difficult to make ordinals describable to the general public. That's because the larger you go the more you simply describe them via logical conditions. For example, a "weakly inaccessible cardinal" is one equal to its own cofinality (shortest possible ordinal-sequence converging to it) and is a limit cardinal (not a successor cardinal). And to describe cofinality, one must describe limits of ordinals, and so on.
back in my day these numbers were big. kids these days with their autologicless+ struxybroken DOS-ungraphable DOS-unbuildable nameless-filkist catascaleless fictoproto-zuxaperdinologisms
We had rkinal-projected number definition with the definition of Aperdinal (Ω∈) isn't FMS-chainable, but can't be RM()^♛/Я^♛-cataattributed to any (cata)thing in Stratasis today
Idk but the idea of inaccessible cardinal seems so fucking badass to me. Been learning bout the continuum hypothesis on youtube to know whether the size of the set of real numbers is Aleph 1 or larger, and the nuance on it is beautiful. Guess this video tackles more on its general idea of larger infinities. Great job!
@@Whybruh-q5bThe Rayo function describes the number after the largest possible number expressed in however many symbols (of first order set theory, whatever that is) the function describes. So, Rayo(10) is the number after the largest number that you can write with 10 symbols. Rayo’s number is Rayo(10^100), or Rayo(Googol).
incredible! this is an AMAZING VIDEO I learned a lot and am glad that the stuff I already knew will be taught to people who don't know it yet, thank you! this is an amazing video that deserves MILLIONS OF VIEWS
if you're wondering what "Fictional Googology" is, it's essentially a version of googology that contains Very ill-defined, if not, completely undefined numbers that should not exist in any possible capacity, which is more of a communal art project about "What if you can count beyond Absolute Infinity" if anything! Even a well-known googologist by the name of TehAarex is in that Community!
I legit did not know tetration was an actual thing! I remember coming up with a very similar concept back in middle school and thinking it was an insane idea. The way I visualized it was "x^x=x2" then "x2^x2=x3", repeat ad infinitum
If you allow things such as "proper classes," then a proper class can be thought of as absolute infinity. However, proper classes don't exist in standard set theory, they can only be reasoned with as propositions instead.
yknow i still wonder who woke up and decided "yknow, what if the 90 degree rotated 8 wasn't the biggest number in the universe?" which caused THIS amount of infinities to be made
"Imagine you're an immortal being floating around in the universe for Aleph Null seconds" *proceeds to make an OC out of this concept and names him Aleph Null*
i feel like nothing can happen after forever, since forever is well, forever. you fill an endless pool with more water, well, you have an endless pool still.
"There’s this emperor, and he asks the shepherd’s boy how many seconds in eternity. And the shepherd’s boy says, ‘There’s this mountain of pure diamond. It takes an hour to climb it and an hour to go around it, and every hundred years a little bird comes and sharpens its beak on the diamond mountain. And when the entire mountain is chiseled away, the first second of eternity will have passed.’ You may think that’s a hell of a long time. Personally, I think that’s a hell of a bird." --The Twelfth Doctor
Actually there's bigger than Gamma Nought: If we use the MDI notation saying that there's nothing bigger by calculating this: {10, - 50,} it can be so big that it reaches gamma. But if use the Gàblën function we can do this: G⁰(0) = 0 G¹(0) = 10^300,000,000,000,000,000,000,003 G² = Aleph null. G³(0) = ε1. G⁴ = Gamma nought... Until we reach GG⁰(0) Or G⁰(1) = I Or incessible Cardinal. So big that nothing in a vacuum is bigger than this. or is it? By using Gàblën function again. We can do GGG⁰(0) Or G⁰(2) = M or Mahlo Cardinal. This is so big that if we use the Veblen function: φ0(0) It would take Epsilon nought zeros to make it. but we can go farther by GGGGGGGGGGG...⁰(0) Or G⁰(10^33) = K or Weakly Compact Cardinal but If we do GGGGGGGGGGG.....⁰(0) or G⁰(ε0) = Ω or ABSOLUTE INFINTY THERES NOTHING AFTER THERES FANMADE NUMBERS AFTER ABSOLUTE INFINTY. ITS SO BIG THAT NO FUNCTION CAN BIGGER THAN THIS BUT JACOBS FUNCTION.
No. The real biggest transfinite number is if you make a function called CALORIES() and put the incomprehensible number, ‘NIKOCADO’ into the function. CALORIES(NIKOCADO) creates a number so big it beats everything else on this video combined very easily, like comparing a million to the millionth power to zero.
good video, however i think it would've been better to continue using analogies relating to supertasks to describe the larger ordinals, rather than talking about "waiting multiple forevers", because that makes conceptually less sense
i love videos like this Very great representation, explenation also with the music! Also writing "The End" in greek letters and aleph 0 was very cool :D
For those of you wondering, the reason Absolute Infinity isn't in this is becaue it's ill-defined (basically there's no real and conventional mathematical definition for it that doesn't create problems) Other than that, great video! I would really like to see an elaboration on Large Cardinals if that's a possibility :D
Just say it encompasses absolutely every cardinal, literally every mathematical expression, even mathematics itself. That's not too hard to comprehend it💀
@@KaijuHDR That's the problem, Absolute Infinity cannot contain everything, if it did, then it would have to contain itself, which makes no sense and causes pardoxes within Mathematics. On the other Hand, if you say "Ω is the set of all Ordinals" there's nothing stopping Ω+1 from existing. Since Ω Is itself another ordinal, thus failing to contain everything.
@@ThePendriveGuy Then what's your point? You just told me it can't be anything then what I just said, which means it can't make sense, which means it ignores all logic. And this isn't even the actualized meaning to it. Cantor just defined it as a infinity larger than everything and cannot be surpassed by anything in everything. Not containing everything. Which don't mistake me saying this, is still probably illogical and paradoxical. Because seemingly it's part of everything, but you also just hinted at the fact that it can't be that ordinal one. Also isnt the "set of all ordinals" just Aleph-null btw? Or another one? I'm too engrossed with making a response (since most of my responses I've reread and realized they're just idiotic and stupid💀) and my own cosmology rn.
@@KaijuHDR My point is, Absolute Infinity isn't a set, or an ordinal, or any mathematical structure for that matter. Absolute Infinity Is better fit as a philosophical Concept, since, like I Said, It causes problems when ported to real math. It's simply something more closely related to the meaning of perfection Cantor also stated himself that it is inconsistent with the definition of a set Also, Aleph-Null Is not an ordinal, nor Is related to Ordinals at all. Aleph-Null Is the set of all counting numbers. While Omega (The "Smallest" infinity) Is simply the thing that comes after all the Naturals. As for set construction, Ordinals and Cardinals are fundamentally defined as sets, so if we invent a new value Larger than any of those, it must be described as a set. TL;DR: Absolute Infinity (Ω) is More of a philosophical concept not meant to make sense in math. It's typically used in your average "0 to Infinity" number videos, which leads people to believe that it is a real number.
The end. 12:24 talking about ψ_1(ω) 14:32 talking about ψ_x(y) 16:37 (heres a rule for this part: y>ωωωωωωωωωωωωωωωωωωωωωωω… [ω times] [or Ω {absolute infinity}]
Well... to be fair.... are infinities really actually definitely larger than each other? In a finite sense, yes. But there is always more infinity, so doesn't that mean that even if one infinity is bigger than another, you can still match every number with another from the "smaller" infinity? Even if the bigger infinity includes every number in the smaller infinity, there are always more numbers. Intuitively it seems that some infinities are smaller than others... But remember the infinite hotel? It depends on how you arrange infinity. Infinity doesn't have a size. It doesn't have an end. If you matched every odd number with all real numbers, they are both the same size. That's because neither of them end. The rate of acceleration is different, but infinity is already endless, no matter what it's made of.
The infinite hotel analogy only works on aleph null many things, because it requires that the collection be countable. That’s how we can prove that e.g. the rationals have the same size as the naturals, because there’s a way of enumerating the rationals that forms a one-to-one mapping between the two sets. However, the argument falls apart for a set like the reals, with cardinality greater than aleph null (maybe it’s aleph 1, nobody is sure), since you can prove that no such enumeration can exist. There are, then, infinities which contain more things than others.
Main reason this isn't true is something analogous to Russel's paradox (in fact Russel's paradox even says some infinities are too large to exist because they result in a logical paradox), comparing a set S with its power set P(S), the set of all subsets of S. Put it in simple terms, there's no mapping f: P(S)→S in such a way that different subsets of S always map to different elements of S, because if such an f existed, then consider the subset B={a∈S | There exists A∈P(S) such that f(A) = a and a ∉ A}. Then consider f(B)=x. Law of the excluded middle says that x∈B or x∉B. In the first case, if x∈B, then by definition of set B, there exists A∈P(S) such that f(A)=x and x∉A. But f maps different subsets of S to different elements and f(A)=f(B), so A must equal B. Which means x∉B, contradicting x∈B. In the second case, if x∉B, then there exists the set B∈P(S) such that f(B)=x and x∉B, so by definition of set B, x∈B, contradicting x∉B. So both x∈B or x∉B are impossible meaning that such a mapping f cannot exist. So any attempt to map P(S) to S must have overlaps, mapping different subsets of S to the same element.
I love this type of video! Keep up the good work ! Where did you learn these things? Did you study it in school or read books independently or did you maybe watch a different video like this? Im just curious:)
A mixture. I first gained interest in infinity from a very old Vsauce video but most of the information comes from books and articles which I read specifically for the purpose of this video.
@@RandomAndgit Recommend reading is the book "Set Theory" by Thomas Jech for more about this subject, in fact it has everything. A pdf can easily be searched for online. However, note that it presumes knowledge about certain subjects, namely prepositional logic (such as what symbols like ∃ "there exists" ∀ "for all"), formal languages, symbols, formulas, and variables and whatnot, basic knowledge about stuff like functions and relations. Later chapters slowly trickle in additional presumptions, like chapter 4 assumes you know about the existence of "least upper bounds" (supremum) in real numbers, and then "metric" "metric topology" "order topology" "lebesgue measure." If you don't know those subjects, chapters 1-3 are still readable and contain the most important basic info, and one can come back to chapter 4 after knowing those other subjects.
Define True Accessor “function” TA: returns smallest ordinal not accessible by its inputs S(x)=x+1 TA(S,0) = ω TA(S,ω)= ω_1 TA(S,ω_1) = ω_2 Make a function out of this TA(S,x)=F1(x) F1^x(0)=A(x)=ω_x A is the basic accessor function TA(A,1) = the inaccessible ordinal at the end of the vid
What on earth is going on in mathematicians brains. This all souns so made up, but I'd be surprised if all those different types of infinities didn't have a rigorous proof behind them that justifies distinguishing them from the others. What a fun video.
When they start adding Latin (English) letters to math 😌 When they start adding Greek letters to math 😕 When they start adding Hebrew letters to math 😱
you should point out the fact that the Infinite stacks of veblen function in a veblen function equals more of a NAN/Infinity relationship, because the Veblen function never gets what it needs in its function slot: A numerical input. It instead always gets a function, which is not able to define the funtion.
Thank you for this amazing video, you explained everything well and thoroughly so that everyone can understand the concept of ordinals, including me! I still have one question after this though: I've never seen an understandable definition of κ-inaccessible cardinals, could you please provide me with one/a link to one?
Sure! I'll try my best. So, k-Inaccessible basically means that a number is strongly inaccessible, meaning that it: -Is uncountable (You couldn't count to it even in an infinite amount of time, for example, you could never count all the decimals between 0 and 1 because you can't even start assuming your doing it in order) -It's not a sum of fewer cardinals than it's own value, basically, you could never reach it from bellow with addition or multiplication unless you'd already defined it. -You can't reach it though power setting (Seeing how many sets you can build with a certain number of elements which gives the same value as 2^x) The basic idea is that you can't possibly reach it from bellow and the only way to get to it is by declaring its existence by a mathematical axiom. Aleph-Null is the best example of something that's kinda similar because it also can't be reached from bellow but aleph null is countable. I hope this helps!
the way I think omega and No is you switch bases like No is the first set of digits and then omega is next like one and tens except with infinate diffrent digits
I think infinity should behave like tetris game. After some point, it will turn negative, then down to zero again. And this point could have been called absolute point since 1/0 equals this point. If we think about the number line is on a sphere, that would make more sense.
I wanted to keep the video within that 10-15 minute mark but I might make a brief followup explaining innacessibles and other even larger ones like 0# and almost huge.
@@RandomAndgit youre gonna need a few parts to explain everything, ciz theres the inaccessibles which you didnt even explain, mahlo cardinals, Inaccessible, weakly compact, indescribable, strongly unfoldable, omega 1 iterable and 0^# exists, ramsey, strongly ramsey, measurable, strong, woodin, superstrong and strongly compact, supercompact, extendible, vopenka's principle, almost huge, huge, superhuge, n-huge, 10-13 and finally 0=1
well, if the Innascesable Ordinal gets reached in the future, we need to then try to reach ABSOLUTE INFINITY, but i dont know if it is fictonal or not.
@@zander513So what is it then? Forever isn’t really an amount of time like a minute is. It’s basically the time equivalent of just infinity. Not aleph null or omega, just the concept of infinity. So, it’s always just forever.
@@Whatdoido-b8c Excellent question. We can actually prove that some infinities are larger or smaller than others using either the powerset or diagonal proof. Essentially, some infinite sets can be matched up to other infinite sets and still have members remaining. For example, the number of fractions is greater than the amount of numbers because you can match each fraction to 1/any number in the set of numbers and then still have lots left over (Like 3/7 which cant be written as 1/x)
@RandomAndgit technically yes, but any infinite number is still infinite, unless there is a tier for transfinities where the infinity we know, is the smallest transfinity
Funny thing is a number named Utter Oblivion is so utterly vast that it is a finite number but surpasses almost all inaccessible cardinals and uncountable infinities
That doesn't work mathematically. Aleph null is, by definition, larger than all finite numbers and all other infinities are, also by definition, at least as large as Aleph null.
@@RandomAndgit If **Utter Oblivion** is a very, very, very large finite number, it would surpass even uncountable infinities in terms of magnitude. This is because its size is constructed to be beyond any typical infinite measure, placing it at a scale larger than any uncountable infinity.
@@RandomAndgit While uncountable infinities describe sizes beyond finite numbers, a number like Utter Oblivion, it is finite and designed to be beyond any typical measure, would exceed even the largest forms of infinity in terms of magnitude.
@@RandomAndgit By definition, Utter Oblivion is intended to be larger than any uncountable infinity. It is designed to be so large that it exceeds the size of infinite sets, including those with uncountable cardinalities.
@@ninas8238 Ahh, I see. Yeah if you're talking magnitude rather than actual size that does kinda make sense. I still don't think I fully understand how that's possible but it could just be me.
I’ve never heard of an eon defined as 1 billion years. Is this different than eons in biology/geology which are defined by fossils becoming different (Hadean, Archaean, Proterozoic, Phanerozoic)?
@@RandomAndgitit's the limit of logic. Every number is a property of it. Even if a number claimed to go larger than absolute infinity, it'll still basically be a property of it.
honestly, anything that comes after omega is can be reduced into a function within itself which can go on forever. kinda unimpressive and ironic because this is an attempt to encapsulate 'forever.'
Notes and corrections:
I mispronounced the atom cesium at the very beginning of the video, pronouncing it 'Kasium'
I said that Omega ^ Omega x Omega is the same as Omega^ Omega ^ Omega when that's actually very wrong.
At 6:11 I used a coefficient with an ordinal when really ordinal multiplication is non-commutative so that could cause problems.
There are several minor phrasing errors around that amounts of alephs and omegas when I'm saying how long to wait.
I had the original idea for this video ages ago when watching a Vsauce about infinity and noticing that it went past many of the ordinals. (Go and watch that video if you haven't, by the way, it's quite a bit more comprehensive than this one.)
Well done! Subscribed!
At 6:10, you momentarily forgot that ordinal multiplication is noncommutative.
@@tomkerruish2982 Oh, right! Sorry. Thanks for pointing that out.
@@RandomAndgiti watched that "powersetting" video of infinity!
6:00 so far this sounds a lot like Vsause’s video
But worth a new subscriber
I can only accept that these concepts were invented by two mathematicians arguing in the playground.
Hilariously, there was actually a real event just like what you described called the big number duel. Mathematicians are just very clever children.
@@RandomAndgitis sams number bigger than utter oblivion or not
@@AbyssalTheDifficultyit’s not a serious number, it’s a joke between googologists
@WTIF2024 Whoa stella, you're in this video?
@@RandomAndgit°-°😮
‘There’s this mountain of pure diamond. It takes an hour to climb it and an hour to go around it, and every hundred years a little bird comes and sharpens its beak on the diamond mountain. And when the entire mountain is chiseled away, the first second of eternity will have passed.’ You may think that’s a hell of a long time. Personally, I think that’s a hell of a bird. (From Doctor Who)
Wow, I may need to watch doctor who.
What season tho ?
@@guotyr2502 season 9
I think that's actually from a story or poem called "the Shephard boy"
the episode is called heaven sent from season 9 if you want to watch it
Ever since vsauce made how to count past infinity 8 years ago, I've wanted to see another video that goes into more detail about the numbers larger than the ones he described, as he jumped almost straight from epsilon to the innacessable cardinals. I've finally found one. This is probably my new favorite video to do with numbers in general.
Wow, thanks very much!
Go check out "Sheafification of g" I'm sure you'll love his videos.
It's rather difficult to make ordinals describable to the general public. That's because the larger you go the more you simply describe them via logical conditions. For example, a "weakly inaccessible cardinal" is one equal to its own cofinality (shortest possible ordinal-sequence converging to it) and is a limit cardinal (not a successor cardinal). And to describe cofinality, one must describe limits of ordinals, and so on.
The sad thing is vsauce didnt explain the cardinals shown at the end in the roadmap and neither did andigit
@@hillabwonSBecause it gets significantly harder to explain
"but there are ways to force past this barrier too!"
me: *"USE MORE GREEK LETTERS!"*
me: "your number plus one!"
@@crumble2000But, on an ordinal scale, +1s don’t matter.
@@MatthewConnellan-xc3oj r/woooosh
this is the kinda content id see from a 100k sub channel
surprised you arent big yet your contents awesome
Thanks so much!
Holy cow I thought you where a big channel until I read this comment! Keep it up dude your content is great
back in my day these numbers were big. kids these days with their autologicless+ struxybroken DOS-ungraphable DOS-unbuildable nameless-filkist catascaleless fictoproto-zuxaperdinologisms
Yet that isn’t even the worst of it 💀
We had rkinal-projected number definition with the definition of Aperdinal (Ω∈) isn't FMS-chainable, but can't be RM()^♛/Я^♛-cataattributed to any (cata)thing in Stratasis today
pretty sure that IS the worst of it
Ik
FG Wiki moment
1:24 I'm sad that you didn't say "this is taking forever"
Damn, I wish I'd thought of that.
@@RandomAndgit what's the biggest number that's not infinite that you can think of?
@@boykisser-1 Good question. There isn't really a largest number I can think of because you can always increase.
Omega is bigger than infinte
@@Chest777YT Yes. That was kind of the point of the video.
You weren't meant to count this high. Turn around
Nah
Idk but the idea of inaccessible cardinal seems so fucking badass to me. Been learning bout the continuum hypothesis on youtube to know whether the size of the set of real numbers is Aleph 1 or larger, and the nuance on it is beautiful. Guess this video tackles more on its general idea of larger infinities.
Great job!
"Hey, are you ready to go on that date we mentioned?"
"Sure, just wait an aleph null seconds."
😢
🤣 or ... are you ready to go out now? just omega seconds darling!
It’s funny just how lightly he uses aleph null like rayo(rayo(rayo(10^100))) isn’t octillons times closer to 0 than to it
@@HYP3RBYT3-p8nWtf is Rayo. I've heard of Tree and Hexation
@@Whybruh-q5bThe Rayo function describes the number after the largest possible number expressed in however many symbols (of first order set theory, whatever that is) the function describes. So, Rayo(10) is the number after the largest number that you can write with 10 symbols. Rayo’s number is Rayo(10^100), or Rayo(Googol).
incredible! this is an AMAZING VIDEO I learned a lot and am glad that the stuff I already knew will be taught to people who don't know it yet, thank you! this is an amazing video that deserves MILLIONS OF VIEWS
i love that all this has no actual realistic use at all lol
this channel has every fact EVER CONFIRMED
You just summoned the entire fictional googology community
if you're wondering what "Fictional Googology" is, it's essentially a version of googology that contains Very ill-defined, if not, completely undefined numbers that should not exist in any possible capacity, which is more of a communal art project about "What if you can count beyond Absolute Infinity" if anything! Even a well-known googologist by the name of TehAarex is in that Community!
@@RealZerenaFando you know if Aarex has a YT?
What a massively underrated channel
I think THIS is my favorite type of TH-cam video. The type that gets you excited to learn about something.
Mine too, I try to make all my videos like that so I'm glad you thought so.
I like how mathematicians attempted making ordinals that can describe Caseoh's weight
lol
it's closer to absolute infinity than anything we know
buccholz ordinal
All muscle, baby!
WHY IS THIS STUPID COMMENT ON A ACTUAL INSTERING VIDEO THE MOST LIKED IM MAD
Another amazing video! Great. I was here before this channel blew up (which I'm sure it will from the quality of content).
Thanks very much!
I’ve watched your videos since the simple history of interesting stuff video, you’ve earned a new subscriber! I really like your content
Infinity so infinite there's infinite infinities, as if it's so infinite that it's infinite.
It's interesting that you take the ordinal approach, i've seen a lot of video that talk about aleph 0 and C, but not so much about aleph 1 ect.
best youtube channel ive ever seen about math so far
damn this channel is underrated af
I legit did not know tetration was an actual thing! I remember coming up with a very similar concept back in middle school and thinking it was an insane idea. The way I visualized it was "x^x=x2" then "x2^x2=x3", repeat ad infinitum
Oh, yeah tetration is really cool. You can do it with finite numbers too, it's part of how you get to Graham's number.
I Like how we showed up to a video about Apierology... I mean, you did summon us, so yay free engagement which means algorithm boost.
Hello There! FG
@@dedifanani8658this person gets it
good job u just did the summoning of all of the fg members
Imagine you said "there is no biggest cardinal!"
But Mathis R.V. said "absolute infinity"
Absolute infinity isn't a cardinal, it transcends cardinals. Also, Absolute infinity is ill defined.
If you allow things such as "proper classes," then a proper class can be thought of as absolute infinity. However, proper classes don't exist in standard set theory, they can only be reasoned with as propositions instead.
@@RandomAndgitwhat about Absolute Infinity - 1?
@@robinpinar9691 surreal ordinals moment
@@robinpinar9691 Absolute Infinity - 1 is still Absolute Infinity.
yknow i still wonder who woke up and decided "yknow, what if the 90 degree rotated 8 wasn't the biggest number in the universe?" which caused THIS amount of infinities to be made
"Imagine you're an immortal being floating around in the universe for Aleph Null seconds"
*proceeds to make an OC out of this concept and names him Aleph Null*
i feel like nothing can happen after forever, since forever is well, forever.
you fill an endless pool with more water, well, you have an endless pool still.
"There’s this emperor, and he asks the shepherd’s boy how many seconds in eternity. And the shepherd’s boy says, ‘There’s this mountain of pure diamond. It takes an hour to climb it and an hour to go around it, and every hundred years a little bird comes and sharpens its beak on the diamond mountain. And when the entire mountain is chiseled away, the first second of eternity will have passed.’ You may think that’s a hell of a long time. Personally, I think that’s a hell of a bird."
--The Twelfth Doctor
One of the best monologue in history of fiction in my opinion.
Can't tell if this killed or fed my infinity anxiety
Por qué no los dos, as they say.
@@RandomAndgitSchroedinger's infinity
Yet it is still closer to zero than…
Caseoh’s weight
This comment is fat shaming
@@jorem_yt its a joke
@@jorem_ytWe love our galactic sized Caseoh
This bends my brain to the point that this whole thing seems ridiculous
Oh wow!!! its me in the thumbnail!
very much enjoyed the TREE(3) reference to your giant numbers video
Actually there's bigger than Gamma Nought:
If we use the MDI notation saying that there's nothing bigger by calculating this: {10, - 50,} it can be so big that it reaches gamma. But if use the Gàblën function we can do this: G⁰(0) = 0 G¹(0) = 10^300,000,000,000,000,000,000,003 G² = Aleph null. G³(0) = ε1. G⁴ = Gamma nought... Until we reach GG⁰(0) Or G⁰(1) = I Or incessible Cardinal. So big that nothing in a vacuum is bigger than this. or is it? By using Gàblën function again. We can do GGG⁰(0) Or G⁰(2) = M or Mahlo Cardinal. This is so big that if we use the Veblen function: φ0(0) It would take Epsilon nought zeros to make it. but we can go farther by GGGGGGGGGGG...⁰(0) Or G⁰(10^33) = K or Weakly Compact Cardinal but If we do GGGGGGGGGGG.....⁰(0) or G⁰(ε0) = Ω or ABSOLUTE INFINTY THERES NOTHING AFTER THERES FANMADE NUMBERS AFTER ABSOLUTE INFINTY. ITS SO BIG THAT NO FUNCTION CAN BIGGER THAN THIS BUT JACOBS FUNCTION.
No. The real biggest transfinite number is if you make a function called CALORIES() and put the incomprehensible number, ‘NIKOCADO’ into the function. CALORIES(NIKOCADO) creates a number so big it beats everything else on this video combined very easily, like comparing a million to the millionth power to zero.
Nikocado is now skinny.
@@w8363 yeah this comment didn't age well
@@w8363It was fake
Calories(Nikocado) is now around 130,000.
I can’t believe Unicode supports all of these symbols
Most of these are just existing letters in greek or hebrew with some subscripts
Clicker Games:
This seems familiar and natural like I've physically been through it before
good video, however i think it would've been better to continue using analogies relating to supertasks to describe the larger ordinals, rather than talking about "waiting multiple forevers", because that makes conceptually less sense
i love videos like this
Very great representation, explenation also with the music!
Also writing "The End" in greek letters and aleph 0 was very cool :D
Thank you very much!
ΤΗΕ ΕΝΔ
Daaaaamn, didn't watch the video but that thumbnail's omega looking dummy thicccc
1:24 missed opportunity to say this is taking forever
Fun fact: everything that is shown in this video is closer to 0 than true infinity
For those of you wondering, the reason Absolute Infinity isn't in this is becaue it's ill-defined (basically there's no real and conventional mathematical definition for it that doesn't create problems)
Other than that, great video! I would really like to see an elaboration on Large Cardinals if that's a possibility :D
It's definitely something I'll make at some point in the future! I'm not sure how long it'll take though.
Just say it encompasses absolutely every cardinal, literally every mathematical expression, even mathematics itself. That's not too hard to comprehend it💀
@@KaijuHDR That's the problem, Absolute Infinity cannot contain everything, if it did, then it would have to contain itself, which makes no sense and causes pardoxes within Mathematics.
On the other Hand, if you say "Ω is the set of all Ordinals" there's nothing stopping Ω+1 from existing. Since Ω Is itself another ordinal, thus failing to contain everything.
@@ThePendriveGuy Then what's your point? You just told me it can't be anything then what I just said, which means it can't make sense, which means it ignores all logic. And this isn't even the actualized meaning to it. Cantor just defined it as a infinity larger than everything and cannot be surpassed by anything in everything. Not containing everything. Which don't mistake me saying this, is still probably illogical and paradoxical. Because seemingly it's part of everything, but you also just hinted at the fact that it can't be that ordinal one. Also isnt the "set of all ordinals" just Aleph-null btw? Or another one? I'm too engrossed with making a response (since most of my responses I've reread and realized they're just idiotic and stupid💀) and my own cosmology rn.
@@KaijuHDR My point is, Absolute Infinity isn't a set, or an ordinal, or any mathematical structure for that matter. Absolute Infinity Is better fit as a philosophical Concept, since, like I Said, It causes problems when ported to real math. It's simply something more closely related to the meaning of perfection
Cantor also stated himself that it is inconsistent with the definition of a set
Also, Aleph-Null Is not an ordinal, nor Is related to Ordinals at all. Aleph-Null Is the set of all counting numbers. While Omega (The "Smallest" infinity) Is simply the thing that comes after all the Naturals.
As for set construction, Ordinals and Cardinals are fundamentally defined as sets, so if we invent a new value Larger than any of those, it must be described as a set.
TL;DR: Absolute Infinity (Ω) is More of a philosophical concept not meant to make sense in math. It's typically used in your average "0 to Infinity" number videos, which leads people to believe that it is a real number.
ω+G looks so cool
underrated channel real
Sorry miss, I can’t attend school today, STUFF, AN ABRIDGED GUIDE TO INTERESTING THINGS JUST UPLOADED!
waiting for the 17 hour video which DOES explain the most complicated functions xd
Some fancy names for infinity, polymorphism of infinity to infinity.
Great video 👍
"Theres no bugger cardinal"
Hey, did you heard of FG? you forgot?
_(It stands for _*_F_*_ ictional _*_G_*_ oogology)_
He's talking about Apierology, where There IS no bigger cardinal, besides absolute infinity.
I never said that there was no bigger cardinal, I just said that it was too big to reach from bellow. (Which is true)
@@RandomAndgitFictional is Fictional¯\_(ツ)_/¯
@@RandomAndgit FG is pseudomathematics anyway
@@theoncomingstorm7903 Quite so.
The end. 12:24
talking about ψ_1(ω) 14:32
talking about ψ_x(y) 16:37 (heres a rule for this part: y>ωωωωωωωωωωωωωωωωωωωωωωω… [ω times] [or Ω {absolute infinity}]
Close your eyes, count to 1; That’s how long forever feels.
Yes, that's Optimistic Nihilism from Kurzgesagt to you blud
That's my favourite Kurzgesagt quote, actually.
so like half a second?
@@BookInBlack hello fellow ewow contestant
agree
I'm interested in the math that you could do with these. I want a sandbox to throw stuff together, like desmos, but infinite.
φχ(0)= is the concept to make infinities higher
Well... to be fair.... are infinities really actually definitely larger than each other? In a finite sense, yes. But there is always more infinity, so doesn't that mean that even if one infinity is bigger than another, you can still match every number with another from the "smaller" infinity? Even if the bigger infinity includes every number in the smaller infinity, there are always more numbers. Intuitively it seems that some infinities are smaller than others... But remember the infinite hotel? It depends on how you arrange infinity. Infinity doesn't have a size. It doesn't have an end. If you matched every odd number with all real numbers, they are both the same size. That's because neither of them end. The rate of acceleration is different, but infinity is already endless, no matter what it's made of.
The infinite hotel analogy only works on aleph null many things, because it requires that the collection be countable. That’s how we can prove that e.g. the rationals have the same size as the naturals, because there’s a way of enumerating the rationals that forms a one-to-one mapping between the two sets. However, the argument falls apart for a set like the reals, with cardinality greater than aleph null (maybe it’s aleph 1, nobody is sure), since you can prove that no such enumeration can exist. There are, then, infinities which contain more things than others.
@@NStripleseven Oh yeah.... that too. Oh well.
Main reason this isn't true is something analogous to Russel's paradox (in fact Russel's paradox even says some infinities are too large to exist because they result in a logical paradox), comparing a set S with its power set P(S), the set of all subsets of S. Put it in simple terms, there's no mapping f: P(S)→S in such a way that different subsets of S always map to different elements of S, because if such an f existed, then consider the subset B={a∈S | There exists A∈P(S) such that f(A) = a and a ∉ A}. Then consider f(B)=x. Law of the excluded middle says that x∈B or x∉B. In the first case, if x∈B, then by definition of set B, there exists A∈P(S) such that f(A)=x and x∉A. But f maps different subsets of S to different elements and f(A)=f(B), so A must equal B. Which means x∉B, contradicting x∈B. In the second case, if x∉B, then there exists the set B∈P(S) such that f(B)=x and x∉B, so by definition of set B, x∈B, contradicting x∉B. So both x∈B or x∉B are impossible meaning that such a mapping f cannot exist. So any attempt to map P(S) to S must have overlaps, mapping different subsets of S to the same element.
I love this type of video! Keep up the good work !
Where did you learn these things? Did you study it in school or read books independently or did you maybe watch a different video like this? Im just curious:)
A mixture. I first gained interest in infinity from a very old Vsauce video but most of the information comes from books and articles which I read specifically for the purpose of this video.
@@RandomAndgit Recommend reading is the book "Set Theory" by Thomas Jech for more about this subject, in fact it has everything. A pdf can easily be searched for online. However, note that it presumes knowledge about certain subjects, namely prepositional logic (such as what symbols like ∃ "there exists" ∀ "for all"), formal languages, symbols, formulas, and variables and whatnot, basic knowledge about stuff like functions and relations. Later chapters slowly trickle in additional presumptions, like chapter 4 assumes you know about the existence of "least upper bounds" (supremum) in real numbers, and then "metric" "metric topology" "order topology" "lebesgue measure." If you don't know those subjects, chapters 1-3 are still readable and contain the most important basic info, and one can come back to chapter 4 after knowing those other subjects.
@@stevenfallinge7149 Ahh, thanks! That sounds like a great read.
Pronunciation of cesium is wild.
They: We have reached another barrier which cant be overcome this time. No matter what!!
Me: what is it?
They : We are out of Greek letters!!!!
I have been summoned: 2:10
Define True Accessor “function” TA: returns smallest ordinal not accessible by its inputs
S(x)=x+1
TA(S,0) = ω
TA(S,ω)= ω_1
TA(S,ω_1) = ω_2
Make a function out of this
TA(S,x)=F1(x)
F1^x(0)=A(x)=ω_x
A is the basic accessor function
TA(A,1) = the inaccessible ordinal at the end of the vid
Is this cheating?
Wait I just realized the final number was a cardinal, not ordinal, eh just replace the omegas with alephs
One of these mathematicians should just announce “Matryoshka’s Number” and call it a day 😂
4:10 The beginning of chaos
What on earth is going on in mathematicians brains. This all souns so made up, but I'd be surprised if all those different types of infinities didn't have a rigorous proof behind them that justifies distinguishing them from the others.
What a fun video.
this is just mathematicians' version of infinty plus one
When they start adding Latin (English) letters to math 😌
When they start adding Greek letters to math 😕
When they start adding Hebrew letters to math 😱
It's actually really simple so uhhh divide zero with zero
you should point out the fact that the Infinite stacks of veblen function in a veblen function equals more of a NAN/Infinity relationship, because the Veblen function never gets what it needs in its function slot: A numerical input. It instead always gets a function, which is not able to define the funtion.
Number is a Endless❤
Is not end yet
You deserve another sub
What if we go the opposite direction... under zero
Really underrated....you can compete with 3b1b at explaining
Wow, thanks very much.
Another great video! Once again I find the music too loud though, you should really consider turning it down
12:07. Nothing impossible for a 📷 cameraman.
rhe kurskazaught intro is crazy
You mean Kurzgesagt?
Thank you for this amazing video, you explained everything well and thoroughly so that everyone can understand the concept of ordinals, including me! I still have one question after this though: I've never seen an understandable definition of κ-inaccessible cardinals, could you please provide me with one/a link to one?
Sure! I'll try my best. So, k-Inaccessible basically means that a number is strongly inaccessible, meaning that it:
-Is uncountable (You couldn't count to it even in an infinite amount of time, for example, you could never count all the decimals between 0 and 1 because you can't even start assuming your doing it in order)
-It's not a sum of fewer cardinals than it's own value, basically, you could never reach it from bellow with addition or multiplication unless you'd already defined it.
-You can't reach it though power setting (Seeing how many sets you can build with a certain number of elements which gives the same value as 2^x)
The basic idea is that you can't possibly reach it from bellow and the only way to get to it is by declaring its existence by a mathematical axiom. Aleph-Null is the best example of something that's kinda similar because it also can't be reached from bellow but aleph null is countable. I hope this helps!
the way I think omega and No is you switch bases like No is the first set of digits and then omega is next like one and tens except
with infinate diffrent digits
I think infinity should behave like tetris game. After some point, it will turn negative, then down to zero again. And this point could have been called absolute point since 1/0 equals this point. If we think about the number line is on a sphere, that would make more sense.
Why can’t it? We kind of just invented all of these numbers for fun anyway.
Its a shame you didnt explain innacessible cardinals tbh
I wanted to keep the video within that 10-15 minute mark but I might make a brief followup explaining innacessibles and other even larger ones like 0# and almost huge.
@@RandomAndgiteventually reaching absolute infinity
@@RandomAndgit youre gonna need a few parts to explain everything, ciz theres the inaccessibles which you didnt even explain, mahlo cardinals, Inaccessible, weakly compact, indescribable, strongly unfoldable, omega 1 iterable and 0^# exists, ramsey, strongly ramsey, measurable, strong, woodin, superstrong and strongly compact, supercompact, extendible, vopenka's principle, almost huge, huge, superhuge, n-huge, 10-13 and finally 0=1
Tbh ω_x is kinda like inaccessible cardinals beta
@@robinpinar9691 by eventually you mean after absolute infinity time?
you sound exactly like the narrator in the old flash game "The I of It". i can't quite put my finger on why
i love the video, but please reduce the volume of the music
well, if the Innascesable Ordinal gets reached in the future, we need to then try to reach ABSOLUTE INFINITY, but i dont know if it is fictonal or not.
btw φ(1,0,0) to φ(1,0,1) is very tricky to look closely
12:21 probably the worst way to write "The End"
Simple answer. Still forever. It's endless and it doesn't stop there. Forever will still be forever after forever.
You just jumped to the conclusion your wrong
@@zander513So what is it then?
Forever isn’t really an amount of time like a minute is. It’s basically the time equivalent of just infinity. Not aleph null or omega, just the concept of infinity. So, it’s always just forever.
0:50 Wouldn’t that make forever finite?
No, actually! It's really weird.
@@RandomAndgit HOW
@@Whatdoido-b8c Excellent question. We can actually prove that some infinities are larger or smaller than others using either the powerset or diagonal proof. Essentially, some infinite sets can be matched up to other infinite sets and still have members remaining. For example, the number of fractions is greater than the amount of numbers because you can match each fraction to 1/any number in the set of numbers and then still have lots left over (Like 3/7 which cant be written as 1/x)
@RandomAndgit technically yes, but any infinite number is still infinite, unless there is a tier for transfinities where the infinity we know, is the smallest transfinity
the universe is 1 forever
12:12 Arent infinities “too big” that we’ve made up numbers?
Funny thing is a number named Utter Oblivion is so utterly vast that it is a finite number but surpasses almost all inaccessible cardinals and uncountable infinities
That doesn't work mathematically. Aleph null is, by definition, larger than all finite numbers and all other infinities are, also by definition, at least as large as Aleph null.
@@RandomAndgit If **Utter Oblivion** is a very, very, very large finite number, it would surpass even uncountable infinities in terms of magnitude. This is because its size is constructed to be beyond any typical infinite measure, placing it at a scale larger than any uncountable infinity.
@@RandomAndgit While uncountable infinities describe sizes beyond finite numbers, a number like Utter Oblivion, it is finite and designed to be beyond any typical measure, would exceed even the largest forms of infinity in terms of magnitude.
@@RandomAndgit By definition, Utter Oblivion is intended to be larger than any uncountable infinity. It is designed to be so large that it exceeds the size of infinite sets, including those with uncountable cardinalities.
@@ninas8238 Ahh, I see. Yeah if you're talking magnitude rather than actual size that does kinda make sense. I still don't think I fully understand how that's possible but it could just be me.
I’ve never heard of an eon defined as 1 billion years. Is this different than eons in biology/geology which are defined by fossils becoming different (Hadean, Archaean, Proterozoic, Phanerozoic)?
Yes, there are a few different eon definitions.
@ so cool! When do people use the billion year version of an eon (btw I just finished the video and I love it)
Aleph null is called countable infinity. ♾️
It's like with washing powder advertising: what comes after whiter than white?
After Forever Is The End Of Math
The definition of WAYTOODANK
the fact he never mentioned absolute infinity is uhhhhhh
Absolute infinity is ill defined and is also neither a cardinal nor an ordinal and, as such, is entirely irrelevant to this video.
@@RandomAndgitit's the limit of logic. Every number is a property of it. Even if a number claimed to go larger than absolute infinity, it'll still basically be a property of it.
@@RandomAndgit but you showed the symbol for Absolute Infinity
@@cuberman5948 No, I showed capital omega which is used both as the symbol of absolute infinity and of omega 1.
honestly, anything that comes after omega is can be reduced into a function within itself which can go on forever. kinda unimpressive and ironic because this is an attempt to encapsulate 'forever.'
Mathematicians love trying to reduce esoteric ideas to functions.
What i got : infinity x infinity infinite times = super infinity, super infinity x super infinity super infinity times = super duper infinity
I mean, that is kinda true. Is it silly? Certainly. Do mathematicians do it anyway? 100%