Things get weird at infinity

[@TheRealGuywithoutaMustache]

I just learned so much, and nothing at the same time

[@zachstar]

Schrodinger's knowledge

[@ςγτε]

Cause you are a bot ! (joking btw)

[@Sasukej2004]

@@ςγτε No he just have a software that show the most watched topics

[@nishanth6403]

You aint just restricted to anime communities arent you?

[@ruatsangawhite7261]

@@zachstar damn that's a good answer

[@Lovuschka]

"Could you store every single picture?" No, the FBI would lock you up for life.

[@stalinjosefstalin480]

That’s true for multiple reasons.

[@Roq-stone]

Lovuschka is that for an infinity?

[@Lovuschka]

@@Roq-stone What do you mean?

[@Roq-stone]

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?

[@Lovuschka]

@@Roq-stone Well, it's because you'd have every possible picture. Meaning you'd also have all possible pictures which are illegal.

[@redsky5495]

zach star is like the substitute teacher you'd have for a class taught by vsauce

[@papazype]

123 321 And that isn’t true

[@rstriker21]

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.

[@redsky5495]

guys, i'm not saying either of them are bad, they just remind me of each other

[@GHOSTrex1324]

Vsauce is god of science videos on yt.

[@azuleno17]

who the hell is that ryan ross?

[@point-xn4tu]

"What happens at infinity?" Well this explanation's going to take forever...

[@shomu_og2841]

... listen here you little sh#t

[@mrsauce9307]

Lmfaoo

[@angelgello3768]

😂😂 This is underrated

[@hassaniq0777]

LMAFAAAAAO

[@future4704]

its gonna take infinity seconds

[@funkyflames7430]

“Size matters... That’s right” I think this is the best thing ever

[@wytzesligting8083]

Your profile pic is Kepler’s supernova

[@goodnightosaka]

@@wytzesligting8083 Your profile pic is Random Access Memories by Daft Punk

[@tuneboyz5634]

Thats what she said

[@MrSirBoastAlot]

@@tuneboyz5634 your profile pic is a detective duck with a hat

[@mosi7486]

@@MrSirBoastAlot your profile pic is Puneet Chaudhary

[@ryunanderson7845]

"Here's a question you didn't ask!" I already knew this was going to be a good one

[@harleyspeedthrust4013]

Classic Zach star comedy, love it every time

[@Thaidory]

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!!!

[@nicolassamanez6590]

not a complex number, that is. wait until you include hyperreals and surreals.

[@Some.username.idk.0]

@@nicolassamanez6590 the uh... the WHAT?

[@NStripleseven]

Nicolas Samanez Excuse me, what? What is that? What?

[@Some.username.idk.0]

@@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?

[@_Nibi]

@@Some.username.idk.0 infinity plus 1 = infinity, but those two infinities are different.

[@boogienightsmarkwahlberg6011]

Zack Star: SIZE MATTERS Also Zack Star: 1 is the same size as infinity. my wife: 😒

[@latneyb]

I have watched so many videos on KZbin 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.

[@comuniunecuosho-campulbudi7611]

John Gabriel New Calculus

[@trueaidooo]

Associative property: *exists* Infinity: We don't do that here

[@angelmendez-rivera351]

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.

[@nothingtodo183]

thats a hi level joke u got there buddy

[@Lawful_Mango]

@@angelmendez-rivera351 I like your funny words magic man

[@adarshmohapatra5058]

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.

[@tszhanglau5747]

(-inf,inf) and (0,1) Zach: they are the same picture

[@ketopp5233]

Cool. That's understandable. Have a nice day

[@stevenjones8575]

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.

[@hamiltonianpathondodecahed5236]

2:03 umm that would be giving tooooo much credits to astronomy Astronomical numbers are much much much ... much less than that monster

[@Flammewar]

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.

[@mitchmcturtle6890]

#nerfgolem

[@woahdude5553]

@@Flammewar uh, did you read the comment wrong?

[@Gamer-uf1kl]

@@Flammewar Hamiltonian Path on dodecahedron says that, that number is way bigger than astronomic

[@SuperDuperPooperScooper4321]

@@Flammewar pretty sure that freak of a number has more digits than there are atoms in the universe

[@charlesjohnson6645]

“Take whatever numbers you want, add or subtract them forever, math happens different. Weird.”

[@michaelmann8800]

That's what you got from this video???

[@funkyflames7430]

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!

[@ttttt_]

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).

[@GordonHugenay]

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.

[@funkyflames7430]

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?

[@theprofessionalfence-sitter]

@@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).

[@GordonHugenay]

@@funkyflames7430 You're assuming too much about this order. When we work with real numbers for instance, we know that if a

[@benwinstanleymusic]

dude i love your videos. the intro with the pictures rly got me hooked straight away. ur the best

[@leoyang1.618]

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.

[@zachstar]

Thank you!

[@TheTelescopescientist]

I agree with this man.

[@XenonX278]

There are over 7100 languages in this world, yet this man decided to speak facts

[@jeremywj]

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.

[@taniamanik2012]

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

[@flowerwithamachinegun2692]

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

[@adarshmohapatra5058]

Here's a video on hilbert's hotel: kzbin.info/www/bejne/i5uWkH6noK5skNE I'm still in high school so this helps me to understand the concept :P

[@parkerhoskovec9400]

@@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.

[@cristianm7097]

@@flowerwithamachinegun2692 Countable = cardinal ALEF ZERO. Omegas are used for ordinals

[@horrificpleasantry9474]

@@flowerwithamachinegun2692 do you care to explain what makes something countable or uncountable, considering no one can realistically count to even one trillion

[@sambachhuber9419]

Good stuff man! I've really been enjoying more of this math content lately. Especially with the presentation of it all

[@AngryPieMan]

I've been waiting for a video on infinity. Thank you, you gentleman and scholar.

[@cb253]

This is one of those videos where I understand less than 5%. But increases my intelligence by 10%.

[@RC32Smiths01]

Interesting. Infinity sure is an interesting concept. I truly wonder if we will ever have the means of achieving such one day. Awesome work!

[@hanniffydinn6019]

You can. Many have. You just take the real life red pills like DMT, LSD, Shrooms. Words literally can’t describe experiencing infinity. 🤯🤯🤯🤯🤯🤯🤯

[@obviouslymatt6452]

Hanniffy Dinn bro you didn’t experience infinitely you took hallucinogenics and hallucinated

[@hanniffydinn6019]

obviously matt You are wrong. Try it and see. God is infinite. You can become the god head. 🤯🤯🤯🤯

[@felipeopazo8375]

We will, in an infinite amount of time

[@friedrichkrone5141]

Dude, humanity reached Infinite stupidity long ago.

[@GuRuGeorge03]

i loved the moment when i first understood this in math classes. nostalgia

[@avi12]

"Can you store every image that was ever created on your computer?" Depends on the compression level

[@nullnull805]

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.

[@sparecreeper1580]

I can only store 9 pixels per picture, and only one picture :,(

[@palmberry5576]

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.

[@dhay3982]

@@nullnull805 That assumes a lossless compression I guess.

[@yosefmacgruber1920]

@@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?

[@brycepowell6639]

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

[@jeffreyjefferson536]

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

[@GreyJedi0916]

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.

[@omeryehezkely3096]

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.

[@scottsaldivar8599]

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.

[@quahntasy]

*Wish we had teachers like zach star*

[@robertschaeffer5861]

Mentor is another alternative..

[@TheRadischen]

But we have... on the internet O.o

[@ashmanideep6253]

Hey, I know you you are a nice animater

[@thom3915]

you always have god

[@yashagarwal8249]

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.

[@malaki1344is]

The infinity between 0 and 1 always blows my mind.

[@tomkerruish2982]

Fun but useless fact: the Cantor set and the Sierpiński gasket (triangle, sieve) have dimensions that are reciprocals of each other.

[@rushunnhfernandes]

Care to explain what these even are?

[@tomkerruish2982]

@@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.

[@skilz8098]

@@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).

[@tonydai782]

@@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.

[@itroll8638]

👍👍👍

[@dragonlord5691]

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...

[@MagicOfDark]

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.

[@samuelallan7452]

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

[@scott-yd3dy]

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.

[@Cirphlyx]

man i wish i had found this for my math courses in college… would’ve made understanding linear algebra and infinity a lot easier

[@jem5636]

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.

[@dawidsienczak9821]

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 ?

[@bayleev7494]

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!

[@bhangrafan4480]

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.

[@conservaliberaltarian2753]

*Possesses every possible picture that can exist -----> *Arrested for child pornography.

[@Prashant-pm7iz]

Wait am I tripping or if you had a computer that could make every possible picture. Wouldnt you make pictures of like anyone naked?

[@MeikiVT]

@@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

[@counterleo]

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”

[@BookofYAH777]

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.

[@yashrawat9409]

1:38 I mean it might as well has the proof for Riemann's Hypoithesis

[@magicmulder]

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.

[@bosnbruce5837]

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.

[@tomkerruish2982]

Papa Flammy said something about a collab with "Sex Star". Do you know anything about that?

[@thomy2562]

Oh, so it wasnt just a fever dream?

[@integralboi2900]

I thought he said ‘sechs star’( sechs is 6 in German), so ‘6 star’.

[@tomkerruish2982]

@@integralboi2900 Ah yes, the old joke. What comes between fear and sex? Fünf!

[@zachstar]

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!

[@NStripleseven]

Lel

[@SollowP]

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.

[@neu8966]

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.

[@lordpersius0322]

Good observation

[@PhoenixTwoFiftySix]

and by the original logic of the question, something would have to be infinite. the pixels, the pictures, the PC, or the storage.

[@gernottiefenbrunner172]

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.

[@Kaboom1212Gaming]

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.

[@counterleo]

@@gernottiefenbrunner172 Well... It only takes 266 qubits to store more information than there are atoms in the universe. We'll get there eventually ;)

[@maypiatt3766]

There is a mistake here at 11:55, this property is only for a function to be injective, but not a requirement for f to be a function. For example, I could give you x and -x, and they would give you the same y value, and x^2 would still be a function.

[@counterleo]

The infective property is referred to as “1-to-1” in the video. The “is also required to be a function” is only here to remind us that one x cannot have multiple images at once.

[@aasyjepale5210]

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

[@futurehistory2110]

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.

[@sandeepunnikrishnan9885]

When I think of infinity Gojo Satoru comes to my mind lol

[@aregulardoggoonaregularday5443]

can we just appreciate this guy teaching us how to calculate the infinite

[@rextransformation7418]

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.

[@enjerth78]

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.

[@MagruderSpoots]

The universe doesn't have enough matter to make that computer.

[@highneedforcognition9660]

How did you conclude the universe is not infinite?

[@Flammewar]

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.

[@highneedforcognition9660]

@@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

[@Mothuzad]

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.

[@Flammewar]

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.

[@Studiacapta-qo2bo]

This is an awesome video! There’s a much simpler (but maybe less elegant) solution to the [0,1)->(0,1) problem: we just need to show there exists an injection from one set into the other, and then by the Schröder-Bernstein theorem they are the same size. For (0,1)->[0,1) the injection is obvious, just the identity. For the other way around, let f: [0,1)->(0,1) be f(x) = (1/3)x + 1/3. Then the boundaries of the codomain are safely out of reach and every element is comfortably mapped.

[@allennelson1987]

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.

[@jenbooob]

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

[@counterleo]

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

[@fireblossom9618]

*Size Matters* *Gets Durex ad*

[@ortherner]

lol

[@accuratejaney8140]

*Size Matters* _gets a Facebook Messenger ad_

[@anshumanagrawal346]

I just came back to this video after almost 2 Years, and realised this is where I started to fall in love with Maths, so thanks for making me fall in love with Maths, Zack

[@sama-tj1rh]

Alright boys you heard it hear pack it up 9:16 :(

[@donovanmahan2901]

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.

[@gaarakashinsaro3911]

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🙏

[@evanw7878]

This literally has 0 applications stay in school kid

[@michaelmann8800]

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.

[@michaelmann8800]

@@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.

[@gaarakashinsaro3911]

@@evanw7878 oh I am pertaining to most of the contents in this channel

[@gaarakashinsaro3911]

@@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.

[@noahrogers7765]

dude you could say literally anything & i’d believe it

[@Mothuzad]

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.

[@skilz8098]

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.

[@RealLifeKyurem]

@@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.

[@skilz8098]

@@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!

[@RealLifeKyurem]

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.

[@sarahbell180]

@@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.

[@einarsalamon8426]

You are messing with my brain at 2 in the morning. I have to go to work in 6 hours

[@nikolaterla5961]

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?

[@GordonHugenay]

The real numbers are strictly bigger than the integers. Every good broad-audience math channel on youtube has made a video about this: Numberphile: kzbin.info/www/bejne/m53ZgI2jZclnfpI Vihart: kzbin.info/www/bejne/aGSsZnqJaa-cesk Infinite Series: kzbin.info/www/bejne/n2jGY6SwbNiCZqs

[@nikolaterla5961]

Oh, I get it. Now my question seems kinda stupid. Thanks a lot!

[@GordonHugenay]

@@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.

[@lostwizard]

@@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.

[@angelmendez-rivera351]

@Joji Joestar No, that is super incorrect. The cardinality of the real numbers is Beth(1), not Aleph(1). Beth(0) = Aleph(0) by definition, and Beth(n + 1) = 2^Beth(n), this is how the Beth numbers are defined. 2^Aleph(0) need not be equal to Aleph(1), and 2^Aleph(1) need not be equal to Aleph(2). The hypothesis that Beth(n) = Aleph(n) is known as the generalized continuum hypothesis. The special case for n = 1 is the continuum hypothesis.

[@radientbeing]

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.).

[@royelhajj2612]

Hey Zach I think you should collab with the gang again (flammy,Andrew,epicmathtime)

[@killerwaspy1303]

This is nice, it's like applying all of my math knowledge to discuss a paradox or a riddle.

[@eddiefirstenberg1000]

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.

[@nightsquill15]

Could you maybe please simplify your explanation a bit😅

[@natvarlalrathod7375]

@@nightsquill15 Just watch the vsauce video. He copied it from there.

[@nightsquill15]

Ok

[@pavlevod]

This channel is gold

[@Josh-oc7ib]

What happens at infinity? You mean what happens when something never ends?

[@chainemusique1792]

The cardinals begins

[@bageda3109]

Well nothing can be infinity so it will never happen

[@Josh-oc7ib]

There are several things that are infinite.

[@bageda3109]

[ Josh ] tell me one thing

[@Josh-oc7ib]

Numbers

[@mewby6494]

This just appeared in my recommend and I don't know why I think I learned something

[@booonnoob7950]

Please make a video for “What is financial engineering. “

[@mr.bulldops7692]

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.

[@mr.bulldops7692]

@ゴゴ Joji Joestar ゴゴ 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?

[@anshumanagrawal346]

@Joji Joestar 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

[@Szymks]

"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.

[@mitchellsteindler]

How would you specify a specific image?

[@alecman95]

Storing isn’t generating

[@Szymks]

@@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.

[@alecman95]

@@Szymks oh wow I wonder why disk manufacturers advertise the storage space if you can just generate everything lmao what

[@bageda3109]

But then you dont got the picture stored. Only the program.

[@darkmatter6878]

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.

[@robertquinn8210]

This is so below basic that I'm embarrassed for Curiosity Stream.

[@Sonicgott]

It is important to understand that infinity is a concept, not a number, which is why logical sequences break when involving infinity.

[@nidhogg6344]

3:23 i don't agree with that statement ... You don't get 0 when you add that up. -1 + a fraction of a fraction of a fraction etc ... you ll have -0.0000000000000_somethingsomewhereinfinityfar ... but not just 0. Each row, will have a higher number. The next one, will be twice as big as the first one. And if you addition them, you ll come close to a -1. So, (-1) + (almost-1) give almost-2 ... rounded up -2. Infinity doesn't break any logic in this. Also, the next example, is not simpler, but way more complicated to understand :D

[@lordspongebobofhousesquare1616]

well mathematically the sum is said to converge to 0. A little different to regular addition, but it's still 0

[@sebbes333]

*@Zach Star* 4:10 That sum doesn't make sense, because this summation doesn't make sense, because a sum should not be directional, it should be a SINGLE & ABSOLUTE number, like 5. To achieve this, you could for example summarize it in a square pattern instead: -1 + 1/2 0 + (-1) ---------------> = -2 + 1/2 = *-1.5* then: (-1) + 1/2 + 1/4 0 + (-1) + 1/2 0 + 0 + (-1) ------------------------> = -3 + (2*1/2) + 1/4 = (-3 + 1) + 1/4 = -2 + 1/4 = *-2.25* And so on... But this result would end up being: *negative infinity* (assuming you start in the top left corner, I assume different starting positions/"corners" give different results?)

[@purifierphoenix2043]

The universe is infinite because it's ever expanding, but since the universe is bound by time, at any moment in time the universe is actually finite.

[@Max_G4]

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.

[@lavihayoun05]

1:10 Since we know that eventually humanity will end, and there won't be any new data to create, That means there is a finite amound of bits that will have existed. which means a large enough amount of storage can account for it

[@ladymecha8718]

1:12 this definition basically explains why AI Diffusion works, it navigates through those finite large group of possibilities.

[@justsomepal6371]

Why are you so underrated. You are the next Vsauce, dude

[@pouria1254]

"You could store every book that can be ever written" So some day we will run out of, creativity?

[@nishanth6403]

3Blue 1Brown and Zach star are the best on YT at explaining math concepts and igniting interst respectively.

[@names9769]

You gotta respect numberphile and mathologer

[@egilsandnes9637]

What happens at infinity ... ... stays at infinity.

[@toaj868]

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).

[@donovanmahan2901]

the comparison of the sets (0, 1) to Z and [0, 1) to (0,1) shows the distinction between cardinal and ordinal values. With cardinals, infinity is infinity. Adding or subtracting 1 won't affect it, but with ordinals, you can say things like "the last element of the set of all reals would be element 2ω^2: the set consists of an infinite set from one integer to the next. From negative infinity to 0 is an infinite set of integers, and from 0 to positive infinity is another infinite set of integers, resulting in 2 infinite sets of infinite sets of elements you would have to go through before reaching the last element.

[@frankyjayhay]

What if, at 14:35, you added 0 to the bottom row. There'd be nothing on the top row to map it to. Then the bottom [0,1) set would be bigger than the top [0,1) set which is supposed to be the same set. Seems strange that it matters when you add the 0, before the proof or afterwards.

[@aviral1337]

woah

[@nahCmeR]

It just looks back around, the second you die in this universe you reborn into another and try again.

[@ayanaxhye]

omg love this video ! your efforts are much appreciated

[@DanielGomez-db4jd]

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.

[@Stoatly4]

The only problem is, it would also store everyone ever doing embarrassing stuff too...

[@butterfacemcgillicutty]

6:10 yea...I'm hopelessly lost...