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.)
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.
‘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)
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
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!
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
"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
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.
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.
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
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
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.
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.
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.
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 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}]
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.
On the point of inaccessible infinities, I prefer the phrasing 'not constructable from the finite.' I've also never seen this topic broached sans the powerset being invoked, was there a reason for that choice ?
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 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
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 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.
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
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.
I'm not particularly fond that this video names so many transfinite ordinals, and yet there is no good discussion on how you typically construct these ordinals. The way it is explained, with supertasks, is kind of loosy-goosey. It's easier and more formally accurate to say that we have generalized the notion of being able to take the smallest finite ordinal greater than any set of finite ordinal (e.g. the ordinal after {2nd, 16th, 13th} is 17th) into sets of any size containing ordinals. This is called the well-ordering property. So the smallest ordinal greater than any finite ordinal is omega, the smallest ordinal greater than omega is omega + 1, and so on.
While that may be true, I feel that the well ordering property is a little more difficult to explain to less mathematically experienced viewers. Thanks for your feedback though.
Construction is actually very simple and easy, you just take the union of all those ordinals. For example, consider 3 = {0, 1, 2}, 4 = {0, 1, 2, 3}, and 5 = {0, 1, 2, 3, 4}, so the union of the set {3 ,4, 5} is the set {0, 1, 2, 3, 4}=5. The union of any set of ordinals is another ordinal, and equal to the sup of that set of ordinals. Of course, one needs to first prove that the union of a set of ordinals is indeed an ordinal.
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.
@@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)
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 know that this is probably a joke but the answer is actually really interesting. So, for any ordinal, we just put +1 on the end (Omega +1, Epsilon0 +1, ect...) but for cardinals we actually change it to its corresponding ordinal +1 so Aleph 42 would become Omega 42 +1. If you do this with an inexcusable cardinal, you can also have an inexcusable ordinal, so that's pretty interesting.
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.'
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5:00 Not technically true. Taking omega squared and appending another omega squared just gives you 2x omega squared. To reach omega cubed you’d need to say something along the lines of “and repeat that process aleph null times”
And also to reach ω^ω you need to say "ω forevers" then repeat that for how many seconds forever is Then You have to repeat That again and again for how many seconds forever is and you get ω^ω
and to reach ε0 you need to do the "And also to reach ω^ω you need to say "ω forevers" then repeat that for how many seconds forever is Then You have to repeat That again and again for how many seconds forever is" Again but this time it's the power of the starting ω and again and again for how many seconds forever is and you get ε0
@@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.
I love how there is one UA-camr that made a number bigger than omega 1 and infinity combined and its absolute infinity that is this Ω also that same UA-camr absolutely destroyed his record with NEVER or ? is there any bigger number than number? Well we can tetrate that with never and make that tower but ive hit a roadblock theres this huge gap between the next number invented by another guy since you can tetrate any number how about NEVER tetrated by NEVER tetrated by NEVER tetrated by NEVER tetrated by NEVER and so on but this number is so gigantic it would probably take 41 eons the number is PROTOCOL BARRIER but whats bigger than it? I mean it's too big to comprehend now imagine every millimeter was PROTOCOL BARRIERs of universes possible we could even just even make our universe inside a parallel universe thats the same as our real universe that has PROTOCOL BARRIERs universes and so on i couldnt even see the number... Edit: the same creator made a bigger number than PROTOCOL BARRIER the end of numbers... Or so i thought the same creator made another number bigger than the end of numbers its B̶̠̤̊Ě̸̢̨̮̙̞͌Y̷̧̘͇̘̿̍ͅO̷̡͕̹̾Ṋ̶̪̬͒̌͒̌̀D̴̝͑ then copying that with the protocol barrier function i just said it comes to the null point... But gigantic numbers can be confusing i mean negative numbers are strange... But can we do the same thing to the null point like the protocol barrier?
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
Actually, the reason I made this video is because of how little that Vsauce video actually covers. He only goes into Omega, Aleph and Epsilon and almost completely skips the ordinals which is why this video focuses on them.
@@RandomAndgit fair enough and the video was of course a very different style than his. its just sad how many times ive seen youtubers make a video like this about ordinals and follow almost the exact same path as vsauce or at least take extreme inspiration and not credit or even mention him. This has to be like the fourth video ive seen do this. I appreciate that hes amazing inspiration and that you added much to the discussion but if you remove his discussion in the intro about 40 and your discussion about eons the first halves of the video follow almost the same plot. Im not saying dont make a video and bring awareness of this awesome part of math to the pubic (also the video you made was really good), im just saying its sad not to credit or mention him for making a video on the topic years ago. I did really appreciate your discussion about the different phi functions and linking googolology tho you definitely added more than others have when making videos on this topic.
Indeed! Well, if you wan't to be really technical, the sun will not explode in the form of a nova, growing into a red giant and swallowing Mercury, Venus and Earth. After another 2-4 eons, the sun is predicted to 'fizzle out' and become a white dwarf. Don't worry though, you'll be long dead before any of that.
"What comes after forever?" "No." The whole point of the video is that there theoretically could be more after infinity which you would know if you actually watched the video.
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
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
‘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
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
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?
"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.
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
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
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?
@@BRAVETOASTA 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.
Yet it is still closer to zero than…
Caseoh’s weight
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.
"Hey, are you ready to go on that date we mentioned?"
"Sure, just wait an aleph null seconds."
😢
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?
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
"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
this channel has every fact EVER CONFIRMED
damn this channel is underrated af
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.
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.
good job u just did the summoning of all of the fg members
What a massively underrated channel
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’ve watched your videos since the simple history of interesting stuff video, you’ve earned a new subscriber! I really like your content
I think THIS is my favorite type of UA-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.
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
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
This bends my brain to the point that this whole thing seems ridiculous
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!!!!
Can't tell if this killed or fed my infinity anxiety
Por qué no los dos, as they say.
@@RandomAndgitSchroedinger's infinity
"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)_/¯
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!
Oh wow!!! its me in the thumbnail!
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
best youtube channel ive ever seen about math so far
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.
You actually can’t count to 6,542,124,659 in 1 lifetime because you can only count to about 100 million in 1 lifetime
No lt 3x10^9
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.
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.
Sorry miss, I can’t attend school today, STUFF, AN ABRIDGED GUIDE TO INTERESTING THINGS JUST UPLOADED!
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!
This seems familiar and natural like I've physically been through it before
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}]
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.
On the point of inaccessible infinities, I prefer the phrasing 'not constructable from the finite.' I've also never seen this topic broached sans the powerset being invoked, was there a reason for that choice ?
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 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
Some fancy names for infinity, polymorphism of infinity to 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.
@@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.
waiting for the 17 hour video which DOES explain the most complicated functions xd
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.
Could you consider turning the music down (or off)? I really struggled to hear and follow you. Thanks.
Sorry! Yeah, a few people have said that. I'm turning the music waaaay down in my next video.
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
12:07. Nothing impossible for a 📷 cameraman.
I have been summoned: 2:10
Another great video! Once again I find the music too loud though, you should really consider turning it down
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.
I'm not particularly fond that this video names so many transfinite ordinals, and yet there is no good discussion on how you typically construct these ordinals. The way it is explained, with supertasks, is kind of loosy-goosey. It's easier and more formally accurate to say that we have generalized the notion of being able to take the smallest finite ordinal greater than any set of finite ordinal (e.g. the ordinal after {2nd, 16th, 13th} is 17th) into sets of any size containing ordinals. This is called the well-ordering property. So the smallest ordinal greater than any finite ordinal is omega, the smallest ordinal greater than omega is omega + 1, and so on.
it's a good critique but keep your personal opinion out of it, you can make a critique without stating your personal opinion abruptly and unkindly.
While that may be true, I feel that the well ordering property is a little more difficult to explain to less mathematically experienced viewers. Thanks for your feedback though.
Construction is actually very simple and easy, you just take the union of all those ordinals. For example, consider 3 = {0, 1, 2}, 4 = {0, 1, 2, 3}, and 5 = {0, 1, 2, 3, 4}, so the union of the set {3 ,4, 5} is the set {0, 1, 2, 3, 4}=5. The union of any set of ordinals is another ordinal, and equal to the sup of that set of ordinals. Of course, one needs to first prove that the union of a set of ordinals is indeed an ordinal.
btw φ(1,0,0) to φ(1,0,1) is very tricky to look closely
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.
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)
9:39 the ackermann ordinal's symbol should be υ (upsilon) since ive never seen it in math
That's not a bad idea, actually. υ could also be good for an ordinal naming scheme after the vebeln function.
υ_α=φ(1,0,0,α) yay
Fun fact: everything that is shown in this video is closer to 0 than true infinity
this is just mathematicians' version of infinty plus one
there is something bigger... its called uncountable infinity. And after that, you have reached the final infinity... ABSOLUTE INFINITY
2019 IS GONE FOR ALEPH-0!
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
Really underrated....you can compete with 3b1b at explaining
Wow, thanks very much.
12:21 probably the worst way to write "The End"
underrated channel real
The only number that comes after absolute infinity is 1/0
rhe kurskazaught intro is crazy
You mean Kurzgesagt?
you sound exactly like the narrator in the old flash game "The I of It". i can't quite put my finger on why
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.
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%
It's like with washing powder advertising: what comes after whiter than white?
Number is a Endless❤
Is not end yet
What comes after CASE-OH’s Weight in Solar Masses
Uploaded on April Fools, 2025
Script:
Start
N O T H I N G
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1
Infinity
Absuolute Infinity
Yeah but what if i add one more
I know that this is probably a joke but the answer is actually really interesting. So, for any ordinal, we just put +1 on the end (Omega +1, Epsilon0 +1, ect...) but for cardinals we actually change it to its corresponding ordinal +1 so Aleph 42 would become Omega 42 +1. If you do this with an inexcusable cardinal, you can also have an inexcusable ordinal, so that's pretty interesting.
Great video 👍
There's known as the average size ordinal call the first uncountable cardinal
Eras of ords ig
Primitive Sequence System Era:0~ε_0
Veblen Era:1~φ(1@(1,0))
Ordinal Collapsing Function Era:1~ψ(Ω_2)
Omega Subscript Era:ψ(Ω_2)~ψ(Ω_ω)
Transfinite Omega Subscript Era:ψ(Ω_ω)~ψ(Λ)
Reflecting Era:ψ(Ω)~ψ(λα.(α2)-Π_0)
Σ1 Stability Era:ψ(λα.(α+1)-Π_0)~(0)(1^4) ig?
BMS Era:0~(0)(1^ω)
Superscript BMS Era:(0)(1)~(0)(1^Ω)
Extended Superscript BMS Era:(0)(1^Ω)~ψ(BFP)
Y sequence Era:?~Y(1,ω)
Meta Absolute Big Omega=Meta Absolute Infinity
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.
{²ω(0)-35,46}=first uncountable cardinal={-ω(1.2,69,29)5}")_}
Let’s just rap this up and say x is the biggest number. As a variable, it can be the highest number.
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Copied and pasted
I know your ass didnt do this
@@caseylynrodriguez342 you dont expect him to write all of that do you
Plus its not related
5:00 Not technically true. Taking omega squared and appending another omega squared just gives you 2x omega squared. To reach omega cubed you’d need to say something along the lines of “and repeat that process aleph null times”
Oh, that's very true actually. My mistake.
@@RandomAndgitand also omega lots of omega forevers just goes to omega cubed not epsilon null
And also to reach ω^ω you need to say "ω forevers" then repeat that for how many seconds forever is
Then You have to repeat That again and again for how many seconds forever is and you get ω^ω
and to reach ε0 you need to do the "And also to reach ω^ω you need to say "ω forevers" then repeat that for how many seconds forever is
Then You have to repeat That again and again for how many seconds forever is" Again but this time it's the power of the starting ω and again and again for how many seconds forever is and you get ε0
The definition of infinity is called absolute infinity, Ω.
Ok but whypanicsec
TELL ME WHYPANICSEC NOW!!!!!!
Parsecs are delicious 😋
I think this type of thing could be called 'the dance of infinities'
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.
I love how there is one UA-camr that made a number bigger than omega 1 and infinity combined and its absolute infinity that is this Ω also that same UA-camr absolutely destroyed his record with NEVER or ? is there any bigger number than number? Well we can tetrate that with never and make that tower but ive hit a roadblock theres this huge gap between the next number invented by another guy since you can tetrate any number how about NEVER tetrated by NEVER tetrated by NEVER tetrated by NEVER tetrated by NEVER and so on but this number is so gigantic it would probably take 41 eons the number is PROTOCOL BARRIER but whats bigger than it? I mean it's too big to comprehend now imagine every millimeter was PROTOCOL BARRIERs of universes possible we could even just even make our universe inside a parallel universe thats the same as our real universe that has PROTOCOL BARRIERs universes and so on i couldnt even see the number...
Edit: the same creator made a bigger number than PROTOCOL BARRIER the end of numbers... Or so i thought the same creator made another number bigger than the end of numbers its B̶̠̤̊Ě̸̢̨̮̙̞͌Y̷̧̘͇̘̿̍ͅO̷̡͕̹̾Ṋ̶̪̬͒̌͒̌̀D̴̝͑ then copying that with the protocol barrier function i just said it comes to the null point... But gigantic numbers can be confusing i mean negative numbers are strange... But can we do the same thing to the null point like the protocol barrier?
Which youtuber is this?
@NO!
(Yes literally He's named NO!)
You deserve another sub
After Forever Is The End Of Math
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?
every one just loves remaking vsauce videos dont they
Actually, the reason I made this video is because of how little that Vsauce video actually covers. He only goes into Omega, Aleph and Epsilon and almost completely skips the ordinals which is why this video focuses on them.
@@RandomAndgit fair enough and the video was of course a very different style than his. its just sad how many times ive seen youtubers make a video like this about ordinals and follow almost the exact same path as vsauce or at least take extreme inspiration and not credit or even mention him. This has to be like the fourth video ive seen do this. I appreciate that hes amazing inspiration and that you added much to the discussion but if you remove his discussion in the intro about 40 and your discussion about eons the first halves of the video follow almost the same plot. Im not saying dont make a video and bring awareness of this awesome part of math to the pubic (also the video you made was really good), im just saying its sad not to credit or mention him for making a video on the topic years ago. I did really appreciate your discussion about the different phi functions and linking googolology tho you definitely added more than others have when making videos on this topic.
@@pr0hobo Crediting is a very good idea actually, thanks. Sorry I didn't think of it.
@@RandomAndgitadd some ordinals between the small veblen ordinal and omega 1
@@RandomAndgit thank you, I think you did a great job presenting the information and i think it complements vsauce’s video quite well actually.
What About Absolute Infinity?
OH MY GOOD
In 5 eons, the sun is going to explode.
Indeed! Well, if you wan't to be really technical, the sun will not explode in the form of a nova, growing into a red giant and swallowing Mercury, Venus and Earth. After another 2-4 eons, the sun is predicted to 'fizzle out' and become a white dwarf. Don't worry though, you'll be long dead before any of that.
ωx_ωx_^ω¹⁰⁰⁰⁰⁰⁰=inaccessible cardinal^ω=mahlo cardinal
forever implies everything to happen ever, if anything came after it then "it" isnt forever, so the answer to the question posed by the title is no
"What comes after forever?"
"No."
The whole point of the video is that there theoretically could be more after infinity which you would know if you actually watched the video.