Things get weird at infinity
2024 ж. 23 Мам.
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Cantor Set Video: • What happens at infini...
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"Could you store every single picture?" No, the FBI would lock you up for life.
That’s true for multiple reasons.
Lovuschka is that for an infinity?
@@BlaqRaq What do you mean?
Lovuschka because the topic was centered around infinity and then you spoke about getting locked up for a lifetime, I just tied in the infinity. You see what I mean?
@@BlaqRaq Well, it's because you'd have every possible picture. Meaning you'd also have all possible pictures which are illegal.
I just learned so much, and nothing at the same time
Schrodinger's knowledge
Cause you are a bot ! (joking btw)
@@user-dk3gv5tm3v No he just have a software that show the most watched topics
You aint just restricted to anime communities arent you?
@@zachstar damn that's a good answer
"What happens at infinity?" Well this explanation's going to take forever...
... listen here you little sh#t
Lmfaoo
😂😂 This is underrated
LMAFAAAAAO
its gonna take infinity seconds
"Here's a question you didn't ask!" I already knew this was going to be a good one
Classic Zach star comedy, love it every time
zach star is like the substitute teacher you'd have for a class taught by vsauce
123 321 And that isn’t true
You ever get a substitute teacher that’s actually good and teaches you something instead of just being a babysitter, that’s what he reminds me of.
guys, i'm not saying either of them are bad, they just remind me of each other
Vsauce is god of science videos on yt.
who the hell is that ryan ross?
“Size matters... That’s right” I think this is the best thing ever
Your profile pic is Kepler’s supernova
@@wytzesligting8083 Your profile pic is Random Access Memories by Daft Punk
Thats what she said
@@tuneboyz5634 your profile pic is a detective duck with a hat
@@MrSirBoastAlot your profile pic is Puneet Chaudhary
Is there a general pattern for an "infinity paradox generator"? 1. Take infinity which is NOT A NUMBER. 2. Put infinity into a mathematical expression where it is treated as a number. 3. ???? 4. PARADOX!!!
not a complex number, that is. wait until you include hyperreals and surreals.
@@nicolassamanez6590 the uh... the WHAT?
Nicolas Samanez Excuse me, what? What is that? What?
@@nicolassamanez6590 so from a quick search I see that this is way too complicated, but I do remember seeing a video ages ago about different kinds of infinities and how some are bigger than others. For example one with some simbol with the index 0,1,2...itself,and then new layers of indexes, is that connected to hyperreals and surreals?
@@sarahbell180 hes a normie don't bother.
Zack Star: SIZE MATTERS Also Zack Star: 1 is the same size as infinity. my wife: 😒
That [0,1) question was in my real analysis class but I never understood the thought process of it so I ended up just memorizing the answer. Only now do I understand how to actually do it
Another way to think of it is in terms of countable and uncountable sets and, more precisely, their cardinals. Basically, all countable sets have cardinal omega, while uncountable sets (like (0,1) or R) have cardinal c (continuum). Now, the way I like to phrase it is that "c swallows omega" as in, adding a countable set to an uncountable set would still give an uncountable set. So, since [0,1) is just (0,1) with one element added, {0}, the cardinal of [0,1) is still c. Note: I did say that uncountable sets have cardinal c, but that's only true for sets like R. You could get sets of higher cardinal by taking the set of subsets of an uncountable set. Sorry for using the word "set" so much
Here's a video on hilbert's hotel: kzhead.info/sun/iM5sj6-pontvjac/bejne.html I'm still in high school so this helps me to understand the concept :P
@@flowerwithamachinegun2692 Omg I remember having to figure out the set of the subset of all real numbers for one of my CS classes. Boy was that a headache to understand.
@@flowerwithamachinegun2692 Countable = cardinal ALEF ZERO. Omegas are used for ordinals
@@flowerwithamachinegun2692 do you care to explain what makes something countable or uncountable, considering no one can realistically count to even one trillion
Associative property: *exists* Infinity: We don't do that here
This is misleading. There is no associative property to speak of, because series are not actually infinite sums, even though they are often presented with historically obsolete notation that visually makes it look like they are infinite sums. They are not an arithmetic function applied to infinitely many terms. Rather, what you have is a sequence, to which you apply a linear transformation, and then you evaluate the limit of that transformation. The transformation involves adding the first n terms of the sequence consecutively, to make another sequence, so in this regard, it is *related* to the topic of sums, but the actual composite operation being carried out is not summation, and so this is why it makes no sense to bring up associativity into the discussion. This is why it is harmful for schools to continue teaching the subject of series as being infinite sums.
thats a hi level joke u got there buddy
@@angelmendez-rivera351 I like your funny words magic man
“Take whatever numbers you want, add or subtract them forever, math happens different. Weird.”
That's what you got from this video???
I have watched so many videos on KZhead which concern themselves with explaining novel concepts in math. I mean everything I can find, from Numberphile to Mathologer to 3 Blue 1 Brown to lectures from university courses at Yale and Stanford to podcasts and interviews with mathematicians and popular STEM educators like Neil Degrasse Tyson and VSauce and Veritasium to presentations at institutions like the Royal Institute and on and on and on.... Yet I find that I grasp concepts and model them in my mind's eye much better after you present them to me. From glimpsing four spatial demonsions for the first time after your demonstration of how a Klein bottle can be made from a cylinder to seeing exactly how conditional convergence can be maniuulated to produce counterintuitive sums, you are so much better at explaining these things than anyone else I have found.
John Gabriel New Calculus
13:25 This reminded me of Hilbert's Infinite Hotel. There you have a hotel with an infinite no. of rooms and every time a guest comes, you make them go to an occupied room. It's occupant is shifted to another room and so on, forever. Here we map 0 to 1/2, 1/2 to 1/4, and so on... . Every 1 / 2^n to 1/ 2^(n+1). The remaining numbers map to themselves. Pretty cool solution actually.
Cool. That's understandable. Have a nice day
(-inf,inf) and (0,1) Zach: they are the same picture
I think it would be helpful to clarify that these rules and definitions for how infinities work are chosen, not discovered. We could define infinity sizes in different ways, for example. Not that these definitions were chosen without reason, mind you; they were chosen because they are useful. Still, I think the, uh, *artificial* nature of Infinities is worth pointing out.
Zach Star is by far the best creator for math content! I love how his videos are so clear and concise, and how there is such a wide range of topics that he covers.
Thank you!
I agree with this man.
There are over 7100 languages in this world, yet this man decided to speak facts
"Can you store every image that was ever created on your computer?" Depends on the compression level
Compressibility is a function of information entropy. One can show by the pigeon hole principle, that for any compression algorithm, if there exists a collection of symbols(a file, including pictures) that can be represented with fewer symbols(its size is reduced), there has to exist a collection of symbols(another file), that can only represented using more symbols using that same compression algorithm. So it is mathematically impossible to represent every file possible, each with fewer symbols. You can see this phenomena in action by trying to compress a zip file twice.
I can only store 9 pixels per picture, and only one picture :,(
Assuming the images are stored as a binary string of pixels with no extra data such as dimensions or camera data, any compression algorithm (that isn’t just the program used to make the images) will only add size.
@@nullnull805 That assumes a lossless compression I guess.
@@dhay3982 Lossy compression would result in many images having duplicates stored. Thus, you violated the condition of storing every possible (digital) image. However, I suspect that the plans for how to build a free-energy machine, would still be readable, so does it really matter?
When I think of infinity Gojo Satoru comes to my mind lol
The thing that instantly popped into my mind was if you created a script that generated every possible picture (even on a smaller scale, say 1-3 megapixel) and had enough processing power and storage to complete it, is that it would be highly illegal. For example, you would have explicit child images of everyone. Images can also be of documents. You would be in possession of highly confidental intelligence reports, tax returns, etc. Of course, you wouldn;t have a way to verify any of them, so maybe no issue there. But the child thing would probably get you a life sentence. Not only pictures btw, videos are just a collection of tons of frames (aka images). You'd have every frame of basically an infiniate amount of child videos.
The main problem with infinity, especially when we consider the cardinality of R (or higher), is that most numbers can't even be stated. i.e., they exist (mathematically), but they can't be described.
2:03 umm that would be giving tooooo much credits to astronomy Astronomical numbers are much much much ... much less than that monster
Nah it’s still really big even in astronomical terms. In our observable universe there are only 10^82 atoms. :D Edit. Sorry, I read your comment wrong.
#nerfgolem
@@Flammewar uh, did you read the comment wrong?
@@Flammewar Hamiltonian Path on dodecahedron says that, that number is way bigger than astronomic
@@Flammewar pretty sure that freak of a number has more digits than there are atoms in the universe
Good stuff man! I've really been enjoying more of this math content lately. Especially with the presentation of it all
i loved the moment when i first understood this in math classes. nostalgia
I had a different line of reasoning for 14:18 Tell me if this is sound but given that [0,1) is a subset of (-inf,+inf) Then the size of [0,1) should equal or be lesser than (-inf,+inf) Given that the size of (-inf,+inf) is equal to the size of (0,1) and (0,1) is a subset of [0,1) and (-inf,+inf), then that must mean that [0,1) is equal to or larger than (0,1), which is also equal to or smaller than (-inf,+inf). (-inf,+inf) >= [0,1) >= (0,1) But (-inf,+inf) = (0,1), so [0,1) must fit in between the two if it is a consistent system. Conclusion: They are all equal in size!
Yes, this is sound and a nice argument. I also present a different version where you take the function f: x-> x/2 +1/4. This function is one-to-one for both [0,1) to (0,1) and (0,1) to [0,1) which implies (0,1) >= [0,1) >= (0,1). Then you already h ave(0,1) = [0,1).
I think the beauty of the proof in the video is that it can be used to show something more general: the union of a finite set and an uncountably infinite set is still uncountably infinite. Outline of proof: 1) Realize the interval (0,1) is uncountably infinite and so we really just need a bijective mapping from (0,1) to (0,1) U {finite set with n elements} 2) In this video Zach mapped 1/2 to 1/4, 1/4 to 1/8, 1/8 to 1/16, etc. These are the reciprocals of powers of two. Notice that 2 is a prime number. You could do the same with powers of 3,5,7,etc. 3) Repeat step 2 n times. You will need n primes. 4) fill in the holes at 1/2,1/3,1/5,...1/p_n with the finite set that has n elements 5) you can verify yourself that this is indeed a bijection. The reason you must have reciprocals of prime powers is to keep it 1 to 1.
It's not trivial that it works. You need to prove that if A is smaller or equally large as B, and B is smaller or equally large as A, then A and B are the same size. If you think that's obvious, then I challenge you to prove it yourself. Just to clarify, when I say A and B are the same size, I mean that there's a function from A to B that is one-to-one and onto. When I say that A is smaller or equally large as B, I mean that there's a function from A to B that is one-to-one (but not necessarily onto). This is always the case when A is a subset of B, just map A to itself.
ElzearYoung It is a contradiction for one to be smaller than the other. If the system is consistent, then a contradiction cannot occur in it. The system is consistent. A = B Do you want another proof that uses a different line of reasoning?
@@funkyflames7430 That isn't actually a proof. You are implicitly assuming that our notion of "being at most as large as" being defined by there being a one-to-one map and "being as large as" being defined by there being a map that is one-to-one and onto behaves like a "normal" ordering, but that is actually the non-trivial part of what you have to prove. Given our definition of size, A not being the same size as B does not necessarily imply that either A is smaller than B, or B is smaller that A (in fact, this being the case is, actually, equivalent to the axiom of choice - if we assume that the axiom of choice is false, there are two sets A and B, such that they are not of the same size, and neither or the two is smaller than the other. Unlike our example, though, neither of those sets is going to be at most as large as the other, either). To prove that A being at most as large as B and B being at most as large A implies them being equally large, what you have to do is show that, given some map A to B that is one-to-one, and another map B to A that is one-to-one, you can construct a map A to B that is both one-to-one and onto (this is known as the Schröder-Bernstein theorem).
Interesting. Infinity sure is an interesting concept. I truly wonder if we will ever have the means of achieving such one day. Awesome work!
You can. Many have. You just take the real life red pills like DMT, LSD, Shrooms. Words literally can’t describe experiencing infinity. 🤯🤯🤯🤯🤯🤯🤯
Hanniffy Dinn bro you didn’t experience infinitely you took hallucinogenics and hallucinated
obviously matt You are wrong. Try it and see. God is infinite. You can become the god head. 🤯🤯🤯🤯
We will, in an infinite amount of time
Dude, humanity reached Infinite stupidity long ago.
Thank you so much for this video! I have been thinking a lot about infinity and this helped solve a lot of problems I had come across but not been able to solve myself.
dude i love your videos. the intro with the pictures rly got me hooked straight away. ur the best
I've been waiting for a video on infinity. Thank you, you gentleman and scholar.
Dangit, what a cliffhanger with the Cantor set! Nicely done, sir. Keep up the great work, I haven't enjoyed maths content this much since binge-watching all of Numberphile :D
I took series class two times (did well) but never fully understood conditionally vs absolutely. I knew how to test it, but didn’t understand why we needed it! Super interesting
For the digital picture analogy, I might suggest Library of Babel. Which is a real site which contains every string of words which can be expressed with lower-case letters, spaces, commas, and periods.
Such Amazing concepts in your videos Always! A good cup of coffee, great for getting mind blown. The Paradoxical concept of approaching Infinity. I love learning about numbers we can use them to mimic reality itself and make good progress. And speculate the inner workings of the next dimension/s.
*Wish we had teachers like zach star*
Mentor is another alternative..
But we have... on the internet O.o
Hey, I know you you are a nice animater
you always have god
These videos are like looking at a hot girl's ass. But actually studying science is like being in a relationship with the hot girl who turns out to be super high maintenance. Trust me I have a Masters.
That last puzzle reminded me of the Hilberts hotel paradox where its a hotel with infinitely many rooms but is fully booked. You can still get someone else to get a room as well, just shift everyone forward to the next room number to book the new person
Yes and if another infinite number of people arrives just make everyone go to their current room number times 2. So card(N) = card Z
This pairs really well with coursework on series, and intuitively understanding why absolute convergence matters. I thought it was just a test and nothing more before this.
Hey Zach, cool video as always! I was surprised not to see the Hilbert's hotel but maybe you already discussed about it in previous videos? If not, personally my favorite part is how to prove that you can welcome an infinite number of buses each with an infinite number of people in them, and create infinitely many free rooms in the process. Cheers,
Thanks! I decided if I'm going to talk about infinity paradoxes like that, I would do it all in one video as its own topic.
*Size Matters* *Gets Durex ad*
lol
*Size Matters* _gets a Facebook Messenger ad_
If you subtract anything from infinity, it is still equal to infinity because you can always add 1 to any number, so their is no number valid below infinity unless infinity is equal to 0.
This just appeared in my recommend and I don't know why I think I learned something
*Possesses every possible picture that can exist -----> *Arrested for child pornography.
Wait am I tripping or if you had a computer that could make every possible picture. Wouldnt you make pictures of like anyone naked?
@@Prashant-pm7iz like he says in the video you would have every image in existence. including a naked photo of every person who exists, or will ever exist
But released on parole and given an infinite number of Nobel prizes for having representations of the equations to cure cancer, end famine, master nuclear fusion and understand the observation barrier and entanglement in quantum mechanics, also P=NP etc. That was before they notice a picture that said “I renounce credit for all discoveries”
I find the first question kind of misleading. Although you specify that you’re talking about 12 megapixel images, the question itself does not. Without it, it seems to me that the answer should be no since there can be pictures that have an infinite number of pixels.
Good observation
and by the original logic of the question, something would have to be infinite. the pixels, the pictures, the PC, or the storage.
You don't even need pictures with infinite number of pixels. Finite but unlimited size still requires infinite storage. Also, you can't actually, even in principle, store all 12 megapixel images, since that is more images than the observable universe has atoms.
Well, a thought process on this shows that there can be infinite pictures and that the video is wrong, in the same way that some infinities are uncountable. Say you have a picture of a forest. Sure. You can then have a picture of me, looking at the computer screen of said forest. To expand further you can have another picture, of another picture of me, looking at another picture of me, looking at a forest. But matter how many times you add on a single picture of me, you can always have another picture of me looking at the previous picture. You can even have more split infinities in top of this by simply adding another picture into the mix. A picture of me, and then a picture of me looking at a monkey, and on a monitor beside a picture of me looking at a forest. You could even nest the previous infinite sequence of me looking at a forest, within the new looking at a picture of a monkey sequence, which is infinite itself. Even with limitations on megapixel size, you can have even more pictures added now, of zooming in to the me looking at pictures of me sequence, like a fractal, spanning down towards infinity and never ending. On top of this, you could have a sequence that includes "every possible image" but then feasibly conceptualize a new image on top of that, of you, looking at a monitor that shows every possible image on it. No matter what, there will always be some other sequence of infinite images that you can create.
@@gernottiefenbrunner172 Well... It only takes 266 qubits to store more information than there are atoms in the universe. We'll get there eventually ;)
Convergent series, you can also have divergent ones too, it depends on the relationship between each term in the series. Not all series are convergent by any means. Just the ones that get closer to a finite limit. NB They never actually REACH that finite limit but you can approach as closely as you care too, just add another term.
omg love this video ! your efforts are much appreciated
man i wish i had found this for my math courses in college… would’ve made understanding linear algebra and infinity a lot easier
1:38 I mean it might as well has the proof for Riemann's Hypoithesis
But also a lot of not quite correct proofs. See this one: en.m.wikipedia.org/wiki/The_Library_of_Babel Yes, it does contain every truth, but finding it isn’t made any easier.
Yesss fractals! I’ve been waiting for this topic for a long time. You guys can go ahead and try Frax HD. It’s a great way to visualize the Julia and Mandelbrot sets.
The infinity between 0 and 1 always blows my mind.
Well I found the images and the -1 + geometric series Matrix to be interesting, and I think it would help if this followed a clear outline.
I'd like to know, how did you come up with that rule applied in the initial grid to form those numbers? I mean, why minus 1 and then sum the rest? Just seems... odd.
You could hold many more of those images by using some simple compression. Example: Say a line of 18 pixels is red, you could store it as red for 18 pixels. Of course this is a very simplified version of compression and our computers so so much more.
No I don’t think so. I believe there is a theorem in information theory wherein any savings you make on nicely compressible images is exactly made up for by losses when compressing the more random images
This is nice, it's like applying all of my math knowledge to discuss a paradox or a riddle.
The universe doesn't have enough matter to make that computer.
How did you conclude the universe is not infinite?
Feelit Believeit If the universe is infinite you could theoretically have an finite amount of matter in the universe. You could measure the acceleration of the universe and calculate the matter. But tbf there are few assumptions to this calculation that could be false. But my point is that theoretically you could have a infinite universe with finite matter.
@@Flammewar yet you could only have an infinite amount of matter with an infinite universe, and you could have a computer of arbitrarily large size with infinite matter
Fun fact: It's trivial to make a computer that could look up any image of size N pixels. You just make a program that coverts an input integer into a different image of that size. 😅 All the data is stored in the lookup addresses that way, but you'd need that many bits in the lookup address anyway, even if you did have a drive containing every possible image.
Feelit Believeit Ok but this isn’t the question. He didn’t ask for arbitrarily big computer instead he wanted a computer which could store every existing picture. For this problem you need a fixed amount of matter, which could be to big to exist in an universe.
Fun but useless fact: the Cantor set and the Sierpiński gasket (triangle, sieve) have dimensions that are reciprocals of each other.
Care to explain what these even are?
@@rushunnhfernandes The Cantor set and Sierpiński gasket (aka triangle, aka sieve) are fractals, mathematical objects which (in a very well-defined way) have a non-integral dimension. Specifically, the Cantor set has dimension ln 2/ln 3 and the Sierpiński gasket has dimension ln 3/ln 2. As for why, here's the reasoning. If we double the scale of a one-dimensional object, its content doubles; for a two-dimensional object, it quadruples, and so forth. The exponent is the object's dimensionality. If we triple the scale of the Cantor set, its content doubles; conversely, if we double the scale of the Sierpiński gasket, its content triples. (This is where images would be soooooo useful.) As for what they look like, well, you'll just need to fire up a search engine. The very concept of non-integral dimensionality is highly counterintuitive, even nonsensical at first, so give yourself time to digest it if this is your first encounter with it.
@@tomkerruish2982 Another analogous way to visualize this is by estimating the distance of a territories coastal line. Depending on how close or far away you are (zoomed in or zoomed out) AND the unit of measure you are using (inches, feet, yards, meters, miles, kilometers, etc...) You will end up with completely different and varying values. Yet the actual size of the coastal line is practically finite (not exactly because it does change over time due to erosion, wind, etc.) but is finite in a given exact moment or frame of time. Yet the dimensionality of these coastal borders has a fractal-like pattern that is not an integer polynomial, they are fractional polynomials. For example, they are not x^2, x^3, ... x^n. They are closer to x^1/2, x^1/3, ..., x^m/n (n != 0).
@@rushunnhfernandes Those shapes are self repeating fractals, there is a way in which fractals can be considered to have a different dimension number than your standard whole numbers. Those two numbers for those two fractals are reciprocals with each other.
👍👍👍
I discovered this many, many, years ago. I call this Pseudo Infinity. What is missing in pictures, is information behind each pixel. As a mathematician I study infinities (such as Surreal Numbers, Ultimate L, Pluralism etc.).
Another way to visualise that any bounded open interval on the reals has the same cardinality as the set of reals is by wrapping the line segment representing said interval around a semicircle. Draw the real number line below it. Now draw lines from the centre of the semicircle passing through both the semicircle and the number line. The point at which each line cuts the semicircle gets mapped to the point where the same line cuts the number line. (It has to be an open interval because the end points go to infinity and negative infinity).
Papa Flammy said something about a collab with "Sex Star". Do you know anything about that?
Oh, so it wasnt just a fever dream?
I thought he said ‘sechs star’( sechs is 6 in German), so ‘6 star’.
@@integralboi2900 Ah yes, the old joke. What comes between fear and sex? Fünf!
Ahhh yes, my alter ego :). I do know something about that though, a little side business collaboration that hopefully will be ready in the near future!
Lel
4:05 earlier you explained that horizontal columns going from right ( 1 ) to left ( 0 ) are just approaching zero which is normal, then you started mixing directions and it got weird , It reminds me of time , time seems to 'flow' one way becouse if you are able to reverse it impossible / paradoxical things would happen thats why it can simply go one way ? it can be slowed down and manipulated by using energy but it can go one way to make certaint sense like in this graph ?
interesting thought, but i thought i'd point out two very important distinctions that can be made. firstly, time has only ever been observed to flow in one dimension, whereas this grid exists in two dimensions. secondly, time is assumed by most well-established theories to exist on a continuum, and the numbers talked about in the video are on a discrete grid. still, interesting idea, and in my opinion you can never be too philosophical about maths!
You are a wonderful person to explain things!
In the set where it's defined (-1, 1/2, 1/4, 1/8, 1/16...) where the sum = 0 across but sum = -2 down is because it's measuring a quadrant of infinity. There is a ceiling and a wall. Take away the ceiling and the sum = 0 for both measurements. Or orient the set so 0 is the corner.
So what are the examples of infiniite sets that are not the same size? Can't we prove one-to-one and onto properties for any two infinite sets by exploiting their infinite nature?
Natural numbers are of the size "aleph-zero" which is the smallest. This is called countable infinity. Real numbers are "aleph-1" which is even higher. You can not biject real numbers to natural numbers. This is uncountable infinity. Let me introduce terminology: The power set of a set A is the set of all possible subsets of A. For example the power set of {1,2,3} would be:{ {1,2,3} , {1,2}, {1,3}, {2,3}, {1}, {2}, {3}, {} } which has 8 elements. The power set of natural numbers is also "aleph-1". The power set of real numbers is "aleph-2" The power set of a set that has the size of "aleph-2" has a size of "aleph-3". So infinities can be as big as we want. The real question is how to interpret anything higher than "aleph-1". We don't even know for sure if there is anything between "aleph-0" and "aleph-1"! That is, a type of infinity that is smaller than the reals but bigger than natural numbers. This is called the continuum hypothesis and is an unsolved problem.
The real numbers are strictly bigger than the integers. Every good broad-audience math channel on youtube has made a video about this: Numberphile: kzhead.info/sun/mNCvf76lZ5Zqe2g/bejne.html Vihart: kzhead.info/sun/ZZeCZauLa3yfd58/bejne.html Infinite Series: kzhead.info/sun/nJucYtWybqWFY4E/bejne.html
Oh, I get it. Now my question seems kinda stupid. Thanks a lot!
@@nikolaterla5961 It's almost always good to ask honest questions, even if they might be stupid. I'm sure someone else had the same question and would have been too afraid to ask, so they'll be grateful to you.
@@nikolaterla5961 That was definitely not a stupid question. Unless you have had a surprising leap of intuition, or reasoning, or have encountered all of this before, it's not unreasonable that you would assume, or at least intuit, the infinite nature of the sets would allow creating a one-to-one onto mapping between any two infinite sets. Basically, infinity weirds everything it touches.
Actually, there are literally more non-terminating decimals between 0 and 1 than there are natural numbers. Allow me to explain: Take an infinite grid, and on the left side write a natural number for each row, with no repeats. For each natural number, add a non-terminating decimal between 0 and 1. Now that the setup is complete, take the first (that isn't the zero) digit of the first decimal and add 1(or subtract 1 if it's a 9). Do that same thing for the second digit of the second decimal, third of third, fourth of fourth, and so on out to infinity. By doing this, you're creating a decimal that is brand new by defining it so that at least one digit is different from each of the other decimals. But, all of the natural numbers are already paired up, so this new decimal makes the total of the nonterminating decimals between 0 and 1 literally bigger than the sum of all natural numbers.
Could you maybe please simplify your explanation a bit😅
@@nightsquill15 Just watch the vsauce video. He copied it from there.
Ok
The "[0,1) vs. (0,1)" is the Infinite Hotel thought experiment! O is just the guest that shows up when all the infinite rooms are full and remapping the 1-to-1 is all the guests moving down one room. Huh. Neat.
That would be more analogous to a countably infinite set and a random number being introduced but yeah
@@hybmnzz2658 ahhhh (0,1) is uncountable. I didn't realize that. I suppose that's why the remapping process is so odd rather than just shifting the mapping?
@@hybmnzz2658 Well, the same thing works if consider the rational numbers between 0 and 1, in the first 0 included, in the second neither included respecticely
I'm studying engineering and yesterday studied progression and series. So this video makes sense to me I like it
3:10 I think it should be noted that the geometric series can never reach 1, so each row adds to 0 minus an infinitesimal; that multiplied by infinity would in this case equal to -2, or if we were precise, it would be the limit
I love math but sometimes I wish my teachers teach us the way you do so that we will know the application of different math lessons... Thank you for teaching us🙏
This literally has 0 applications stay in school kid
The applications of certain mathematics lessons can be varied, and some of them can't be made apparent to you because you don't have enough knowledge of mathematics to understand the application (take, for example, "imaginary" numbers, which I am constantly told make absolutely no sense by people, mostly because they get hung up on the word "imaginary" and won't let go). That doesn't mean you can't learn the rules and use the rules for simpler things even though you don't fully understand why the rules are that way or what their grander applications are. If the advancement of human knowledge REQUIRED that people always knew what the "applications" were beforehand or even at the same time, we would never learn new things and never have access to new applications. Nothing would develop. You are thinking about learning in a completely backwards way. It may make sense to you to do that, but it is ultimately going to get in your way.
@@evanw7878 The first part of your comment is literally 100% incorrect. "Applications" doesn't mean something the layperson uses in everyday life. An entire branch of mathematics deals with stuff like this, and that branch of mathematics most certainly IS used in other areas. Stop feeding such lies to people.
@@evanw7878 oh I am pertaining to most of the contents in this channel
@@michaelmann8800 Very well said. The problem is that most of my teachers doesn't even know how some formulae are constructed that way. They just make us memorize the formula and give us same format of questions which doesn't do us any good especially if we are given a very different format of question from the previous ones that they gave.
This channel is gold
About the picute question: Technically, this could be made without using that much storage. For example, there is a website that contains any text of a certain size (I think it's 2400 characters or something like that) that could be written using the english alphabet, dots and commas. Tje site's name is Library of Babel. It doesn't store those texts, rather it generates them based on a Hash. However, each combination always results in the same result. You can explore it or you can search for a specific text or part of texts. Technically it can be made with pictures too.
Hey Zach I think you should collab with the gang again (flammy,Andrew,epicmathtime)
Alright boys you heard it hear pack it up 9:16 :(
3:30 Riemann series theorem ( rearrangement theorem) : If the sht converges but not absolutely, then it can be rearranged in a permutation so that the new series converges to an arbitrary number, or is divergent.
Could you store every picture that exsist wow to the questioning. Terrill TC!
Please make a video for “What is financial engineering. “
What happens at infinity? You mean what happens when something never ends?
The cardinals begins
Well nothing can be infinity so it will never happen
There are several things that are infinite.
[ Josh ] tell me one thing
Numbers
2:43 this is where limits come in, you start with a size of 1, then 2, then 4, then 821575793423, until you reach a number that approaches infinity that lets you estimate the limit. It's -2 in this case.
Great video! Thank you very much for your effort.
It’s wierd to think that there couldn’t be an infinite number of of variation in humans if we ignore microscopical levels
You could say it for life in general, there is a finite number of how a being can looks like, it huuuuuge but finite
Yes I do get that but takin it down to smaller things helps giving it a more handleable perspective
But yeah your right
Fun fact: The set of all computable numbers is countable, whereas the set of uncomputable numbers is uncountable. This means that the real numbers are uncountable only because they contain uncomputable numbers! So whenever someone thinks they've found a way to enumerate all real numbers, they're just enumerating computable numbers, and missing what makes the reals uncountable in the first place.
Not exactly true... in cartesian plotting yes this could be determined, however, if we expand this to the complex numbers and plot them in the complex plane using polar coordinates... you can in theory map every real. An example of this is taking the roots of a quadratic... when we map them in cartesian space where the parabola is either above the x-axis or has one point tangent to the x-axis we end up with either 1 or 2 imaginary or complex roots and we can not graph or plot them. However, if we expand the cartesian x-y plane to include the complex plane we can then map every root.
@@skilz8098 The complex plane is larger than the real plane. In fact |ℂ| = |ℝ²|. In other words, the cardinality of the complex plane equals the cardinality of the real line squared. We don't know any uncomputable number because we can't compute it. Therefore we don't have a representation for it nor can we know its value. You're proposing an algorithm, in other words, a way to compute the number. Therefore you're still operating within the set of computable numbers, not the real numbers.
@@RealLifeKyurem I wasn't exactly suggesting that... basically just because we don't know how to compute it, doesn't mean that it can't be!
skilz8098 There are uncomputable numbers, and it’s not a matter of knowing the algorithm. Let ψ be an uncomputable number. By DEFINITION, there is no formula, mapping, nor algorithm that can tell us what it’s value is. If there is an algorithm, even an unknown one, it’s not an uncomputable number anymore.
@@RealLifeKyurem Actually, there are as many complex numbers as there are real numbers in a set theoretic sense, though it requires first showing |ℝxℝ|=|ℝ|. A clever way of doing that is this. Since the cardinality of ℝ = |(0,1)| = |(0,1]|, we can freely play with decimals without whole numbers in front. For a decimal expansion of two numbers x and y ( (x,y) an element of (0,1] x (0,1]), there are nonzero numbers by which we can segment each decimal into blocks. Now, for each real in (0,1], there is a nonterminating decimal expansion (for instance, 0.1 can be represented as 0.0999...). Then, segment each decimal based on when the ending is nonzero. So, for say 0.3510031903..., we get the blocks 3 5 1 003 1 9 03. And for say 0.29031053006... we get 2 9 03 1 05 3 006. Then, doing this for x and y, we can form a new number in (0,1] as follows-start with the first block of x, then the first block of y, then continuing on to the second. We get the decimal .325910300311059303006... By this process, one can decompose by this alternating method each number in (0,1] as a block pair and form two block pairs that correspond to (x,y) in (0,1] x (0,1], so it is one to one and onto.
If you want it, you can use this for adding infinity numbers U0 x 1 / (1-r) U0 is the beginning And r is the multiplication, Only works if r is smaller then 1 and bigger than 1 So if you do x 1/2 and start with 1 its 1x 1/1-0,5 and the answer is 2... If you start with -1 -1x 1/1-0,5 and the answer is -2...
The only problem is, it would also store everyone ever doing embarrassing stuff too...
6:30 Its still not valid, you can't just go about picking the most comfortable way to sum up, that should be your point.
Actually you can. You see the ambiguity is in the "..." we use at the end of an infinite sum. It does not do justice to associativity which is the true problem of Riemann Rearrangement theorem. In sigma notation it is more clear what is happening and you can choose to construct anything. Remember that infinite sums are defined as limits of finite partial sums.
Hassan Akhtar That last sentence is often what clears up infinity for most. We have to define infinity and explain what we mean by it so that people can actually understand its implications.
Here is something unrelated to the rearrangement theorem but relevant to associativity and infinite sums: What is 1 - 1 + 1 - 1 + 1 - 1 + ... ? A common beginner answer is "it depends on where you put brackets". (1-1) + (1-1) +... = 0 (1) + (-1+1) + (-1+1) +... = 1 (1) + (-1) + (1) + (-1) +... = NOT EXISTS Of course we standardly assume the answer is does not exist because we think the partial sums are defined term by term. In reality, sigma notation removes ambiguity and all 3 of those results are valid as long as you define your sequence. That is how we deal with associativity. For example what if you want 1-1+1-1+.. to be equal to 1? Let a_1 = 1, let a_n = (-1+1) for n>=2. Then the infinite sum from n=1 to n=infinity of a_n is indeed 1 (again think of limits of partial sums). This is effectively like putting brackets like I did above!
Hassan Akhtar Keep on teaching and preaching the word of math! I love it
It is valid. For conditionally convergent series you have one sum for each ordering of the terms, and for absolutely convergent you have exactly one sum for any ordering.
My donger is the limit.
bruh
Bromine Uranium Hydrogen
nah, negative infinity _it goes backwards_
Smaller than Epsilon?
@@zachstar :O
Brilliantly made
Thanks for the video.
This is so below basic that I'm embarrassed for Curiosity Stream.
He made a mistake when he was explaining the table with the negative ones. He made a diagonal line near the bottom left -1 meaning that there is a third option.
There are more uncountable numbers in the real set, than countable. The set of uncountable numbers is even so big in comparison to the countable ones, that the probability of choosing a random real countable number is 0
Ask anyone on the street to give you any real number and I’m pretty sure they’ll choose an integer with nonzero, potentially .5+ probability 😀
You forget something for movies. The amount of ways to combine these pictures into a temporal sequence and encode that sequence is totally intractable, even though the number of pixels is finite in each picture.
"Could you store every single picture on a computer?" Just write a simple script that generates a random image. You can now generate every single picture that can exist.
How would you specify a specific image?
Storing isn’t generating
@@alecman95 In computers generating and storing is actually the same thing. You don't store a picture, you just store the instructions to generate the picture.
@@Szymks oh wow I wonder why disk manufacturers advertise the storage space if you can just generate everything lmao what
But then you dont got the picture stored. Only the program.
1:19 I've a question. Based on the fact that all photographs involve said number of pixels as shown here, does that mean the only thing preventing 100% real CGI generated movies of well anything (even CGI generated drama and comedy movies made to look like they were filmed IRL) is simply down to limitations of computer processing and the time needed to generate such detailed imagery for some 150,000 frames in an average movie? Either way, it shows how in the future this should be entirely possible, maybe in 100 years.
in 4:30, the sum of rows isn't 0. It's -2. If you say it's zero just because each element is approximately 0, then you might as well say that an integral is always zero just because each infinitesimal element is approximately zero. If you sum up the infinite number of infinitesimal negative values of the rows in the infinite matrix, The sum will still be -2.
I love numbers thanks for making this video
Love the Dark Tower picture snuck in there :)
can we just appreciate this guy teaching us how to calculate the infinite
Came up randomly on my feed. Somehow I understand it all but can't regurgitate a thing 😭 I love it's chaos 🥰
I'm currently studying Computer Science and I learned about this stuff in Discrete Math. Very interesting stuff and actually makes sense when it finally clicks. Science is awesome.
I learn more here and at home than in school
This is one of those videos where I understand less than 5%. But increases my intelligence by 10%.
To get a better grasp of what infinity is in a finite situation. Try to find the beginning of an already made circle. where does a circle start and where does a circle end? or a sphere if you rather prefer that.
There arent many easily understood set theory overviews online nice