An Infinity Paradox - How Many Balls Are In The Vase? 

How can you CHOOSE the number of balls left in the vase at noon? HINT BELOW *What pairing are you using?

“This not an infinite number of balls, I’m on a budget” The budget’s the only problem? SHE KNOWS SOMETHING WE DON’T 😂

I'm envious that your alter ego is called BLADE

"Cow-nting" Ha!

"let's assume you forgot how to count" that was my day at work.

I heard this paradox before, but when you drew the relation to 1:1 pairing, that helped me understand it so much.

Its too early in the morning for my brain to be hurting this much. I love this channel

Well, it took a while but I did the maths, came up with 42.

Occasionally the RU-vid algorithm will throw you a gem … this is one of those times. Fantastic Channel! … and years’ worth of content to catch up on.

Hi Jade, This video blew my mind as virtually all your wonderful videos do. When l was midway through this part video, the Mandelbrot Set (MBS) came to my mind.... I thought, "wouldn't it be great if Jade did a video on that?" Then near the end, my mind was blown again when you actually showed a colorized MBS! I wonder if you've done a video on the MBS? If so, l'd LOVE to see it. If not, might you consider making one on this amazing phenomenon? P. S. THANK YOU for all that you do 🎉

This is a variation of the Hilbert’s Hotel paradox. Both illustrate why a physical infinity is an impossible concept.

I just imagine the store clerk ringing 20 ball containers and judging you silently

2:21 Blade appearing out of nowhere and just saying "my vase is empty" lol

4:00 better not do this part during a pandemic. That many milkshakes will SURELY bring all the boys to the yard.

"If we treat time as infinitely divisible" Discrete Time Theorists: "Fools!"

When I started in aeronautical engineering and math, I had lots of trouble with infinity. At 76, infinity has no fears for me. Maybe I have unconsciously changed from being an engineer to a philosopher as I got older. How aircraft fly is still magical and how they can get a drone to fly on Mars is even more mystical.

I love that animation of the ball rolling around the Penrose triangle!

3:13 Yeah, that's my facial expression for every paradox I've heard about...

Hey! Did you sneak in and use my cloning machine without asking? 🤔 (Also, good video.)

There is a confusing mistake in the beginning of the video. You meant to say take out balls 1, 2, 3 NOT balls 10, 20, 30.

I would suggest there's a contradiction in the original paradox in that you're directly linking an infinitely divisible continuous variable (the time stamp) with an indivisible discrete variable (the number of identical balls). Hence you get a contradiction to which either answer gives an incomplete answer.

I like how you showed the infinity symbol and an impossible triangle side by side. This is exactly what came to our mind when me and my friends designed a logo for our indie game studio, our slogan was "Everything out of Nothing", we wanted a zero and an infinity, we ended up drawing a rectangular zero that is also an impossible shape , then evolved it to a round zero that it also an impossible shape (kind of like a mobius strip).

11:28 Not many cosmologists really believes anymore that the universe will do the "big crunch" - on the contrary: vast majority believe rather the heat death or "big rip"...

10:33 I guess everybody here asked this question to himself , well, when I asked this question to myself, when I was younger, my brain said 1 million and I be like okay👍

Huh, I was thinking you were going to talk about the Doomsday Argument but I guess we can save the existential crisis for another day, just some good old fashioned Up and Atom brain melting!

I agree completely - as a phi nerd and a math nerd, I am positively _fascinated_ by the intersection between them. Many of my friends roll their eyes in a "there he goes again" look when I start talking about this :-)

If time is infinitely divisible, you would never reach noon in this thought experiment.

This video reminded me of the fact that a conditionally convergent series can be rearranged such that their sum equals any number! Wonder if they are related.

The concept of infinity is fascinating I always had the thought in my mind that if infinity and infinitesimal behave differently rules, why is the world so predictable? This question made me question myself so much I took physics at my university. Eventually, I realized infinity and infinitesimal in their full glory are just concepts and do not physically contribute to the world[I am pretty sure I framed this sentence so bad I seem a lunatic now]. Like we cannot have time smaller than Planck's time, cannot have speed more than the speed of light, cannot have accuracy beyond the Heisenberg principle. Every place where one could imagine bringing an infinite count or an infinitesimal count/accuracy, there is some rule of physics waiting to disappoint your idea. To me, this seems to make the world predictable. I mean if the behavior of infinitesimal silicon atoms(yes, they are not. I'm just saying if they were) were different from time to time, I would not be sure if this was the exact message you were seeing. If neurons behaved at infinitesimal accuracy, I would not be sure I could control my feelings in front of my crush. And while some say it would have been great if there was one physical non-barrier to infinity; the idea that things are normalized, countable, quantized, discrete, understandable, fascinating really helps me sleep at night.

"I'm on a budget" 😆

1:01 Infinite balls because we never reach noon

This video is the best reason for me to get curiositystream so I can watch your bonus videos on Nebula

"I'm on a budget" Lol I loved that

Aren’t Invention Vs discovery, philosophy Vs mathematics, infinite Vs finite questions of reasoning and hypothesis based on the media of natural languages? If we choose another medium to communicate might we not discover the presence of multiverses as well as the causal and non causal ? With such knowledge, Jade, won’t we be silent in awe and wonderment or keep counting and calculating?

It's still (+10n-n)=(+9n) at each step, so it doesn't really matter how many steps you take, the final result is still 9n, which is +infinite at the limit. As you said, what the second method does is creating a 1:1 correspondence between the - removed - balls and the number of time steps. The second method has completely forgone counting balls, even if it could have been possible to do it, exactly as it has been done in the first one by counting 9 balls for each step. In the end, the two methods are exactly the same: the only difference is our attention is focused on two different things.

"Did I ever tell you the definition of Infinity?" -Vase

8:45 “We were both right” That means 0=∞ and whole number line collapses to one single point

Surely the reason it seems so bizarre the way infinity ♾ behaves is because in both cases an infinity could never form, interesting thought experiment, but thinking about it will obviously give paradoxes because treating it like a number is like dividing by 0 (if you haven’t seen there’s something interesting proofs if you can divide by zero that leads you to be able to say that any number is equal to any other number), anyway let’s ignore that for a second, surely in both cases you would get an infinite amount of balls both in and out the hoop because again infinity is not a number so don’t treat it as such(more of a concept that’s useful to use), so the only way to “solve” it would be to think of the problem logically, for example when you see an asymptote on a graph, to get an idea of the value it’s converging on you repeatedly choose answers which get closer to seeing the value the asymptote is at and make an approximation to see what it converges at (because we can’t actually repeat this to infinity) but we know this is pretty accurate to how the infinity would work

Paradox: same setup, but each step you put in 10 and take out 9

"My milkshake brings all cows to the yard / I can teach you, and there is no charge"

4:09 Obviously there would be infinite balls both inside and outside the vase but lemme explain. An infinite number of balls are put in the vase in sets of ten and for each set, one ball is taken back out. Since more were left in than taken out, there are still infinite balls remaining inside the vase. But since one ball was removed each time ten were added and this was done an infinite number of times, that obviously means there are also still infinite balls on the outside too. Shouldn't that be basic math? I don't understand how that could have been considered a paradox.

Reminds me of series that can converge to any number depending on how you arrange the terms.

- Just sharing my idea at 5:36 so haven't watched the "fun" part yet - I don't see a paradox here. In the second case, you have infinitely many balls too. Of course you can tell when a specific ball is taken out of the vase, but I can't reasonably specify every one of the infinitely many balls. The first ball that stays in the vase already has infinity written on it (you probably lost count I guess), and the last one too. I mean it's kinda like saying there are no sharks in that sea, and trying to prove it with the fact that I don't even know what a shark is. But let me also go one step back, because in fact, there is no way you have no balls left in your vase. For every timestamp, you take out one, and put in 10. If we are checking at noon, then we are (somehow) over with the throwing balls around, so there was a LAST timestamp, when you put in 10 balls, and took out one. It doesn't matter what was the ball count before, given that you can't have negative amount of balls, you are always going to have at least 9 balls in the vase.

This seems about as stupid as "You observe one person enter a building and then two exit. If one more person enters the building then it will be empty (1-2+1=0)". The two sets might have some kind of equivalence but the way the problem is described there is no point in time ever where there are 0 balls in the vase. It is always growing faster than it is shrinking. Either your math is broken or it's useless for modeling anything in the physical world. It seems that your choice of applying set theory to figure out how to model this related rates problem at infinity is not a good one, and it hints that perhaps not all infinities are equivalent.

Challenge Question: You are in a starship in orbit around the sun. You desire gravity but spinning makes you sick . You decide to decrease your orbital speed until you feel a 1g pull from the sun. To maintain your distance and not crash into the sun as you slow down you use a giant solar sail to to provide the needed "lift". Where would your balance point be? How big a sail would be needed for your ship? I assume you could generate a graph of safe speeds/sail size to give you a region of conditions that you would feel normal gravity toward the sun. Or in other words, is it possible to balance on beams of light to keep from being vaporized in a fiery death?

This is how the government gets your money, yet always seems to increases the national debt.

What I will say, though, is regarding infinity, the Mandelbrot Set illustrates infinity in a finite space. So that’s pretty cool. Love you, Jade. Never stop posting, please.

How log table is created?(plz, make a video on this topic)

I was right , it's about infinity 😃😃

With an infinite volume vase and infinite balls getting filled, at any given point of time, is the vase 100% full or 100% empty?

Can we divide infinity into types? 1) Infinity extending on one side (eg- 1,2,3,4,5,...) 2) Infinity extending on two sides (eg- ...-3, -2, -1,0,1,2,3,... ) 3) Infinite things between 2 things in an infinite line of things (eg- {Set of Rational numbers} or {Real Numbers}

The problem comes from applying the word 'all' in 'pairing all elements of an infinite set'. 'All' or 'every' become semantically nonsensical when applied to infinity because you can't exhaust infinity.

2:58 What about ball 3,000,436? Is that still in there? Well no, that was taken out on the 3,000,436th step... when I added balls: 30,004,351; 30,004,352; 30,004,353; 30,004,354; 30,004,355; 30,004,356; 30,004,357; 30,004,358; 30,004,359; 30,004,360..... Well weren't all 10 of those balls taken out? Well yeah, when I added 90 more because there's a 1:1:9 correspondence of [steps] to [balls removed] to [balls added] since every ball you can name has a corresponding step removing it AND 9 more corresponding balls YOU CAN ALSO NAME. With complete sincerity, I love ALL of your videos and I really appreciate your contribution to the internet as a whole. I just have a pet peeve when it comes to explaining infinity because despite the fact that infinity is mathematically sound, -attempting- to rationalize it with mathematical logic just does not work because MATH does not work on infinity. LOGIC does not work on infinity. It's literally incompatible. You can't add, subtract, divide or multiply infinity; You also can't reason with it, and It can't be manipulated in any way. I personally believe infinity can't be taught.... traditionally. Rather than speaking about what it is, I feel the only way to really talk about it is to talk about all the ways infinity DOES'NT interact with math. It's like how black holes interact with physics :D

Hey, In your video about Schrödingers cat you wore a t-shirt with an elephant on it. I always draw those kind of elephants on everything because it is the only thing I can draw. I would love to get that t-shirt for myself so where can I buy it?

“any number of balls can be left in the vase at noon” idk about that, good luck leaving π balls in the vase at noon.

It's not a paradox. The answer to the first question is infinity - 1. The answer to the secondary argument, where she just sequentially removed the balls, starting at 1,2,3 etc.... is still infinity - 1. The balls that remain always outnumber the balls that were removed, infinitely, due to the initial equation. To help you better understand why pairing the sets is irrelevant, just think of the initial visual she created in the video, and wipe the numbers off the balls. The pairing, the "label" of each number is technically irrelevant. It's has no more meaning than the number being a co-ordinate, but it doesn't give each individual number more or less value. a single ball is a single ball, be it the first, or the billionth in a set. This is what happens when you start a thought experiment, that is quickly nullified, once you introduce "and divide it by 2, infinitely"

0:22 ''I'm on a budget'', well if you ask Elon Musk he will say the same thing

I think that there's just lack of information on the procedure. Clearly which ball gets removed aftere 10 are added matters. Keep also mind, that we both add and remove infinitely many balls and infinity minus infinity is indeterminate, so I'm not really surprised that the answer could be anything. This doesn't seem too hard to imagine that we still get something reasonable depending on the adding and removing procedure. What impresses me more is the Thompson lamp paradox. In that case, it doesn't matter how you look at it the answer seems always wrong, suggesting that not all super tasks can be possible to conceptualize.

You like paradoxes and I think that contagious cause I have started thinking about paradoxes. Should I drop RU-vid premium and get curiosity stream? Streaming is the paradox of cable.

Not a philosopher. Not a physicist. I live on the street and I'm very happy. I remember realizing that all things in the universe are on the cusp of two infinities. The infinitely small and the infinitely large. So there you are. Another paradox. And Jade, can't afford Nebula but love you on RU-vid

I think the vase can not be empty because when u do 1 step u add 10 and remove 1 so we have 9 and for 4000000th time we add 10 and remove the 4000000th so there are still 9 more balls.

My favorite version: Step 1, add balls 1-9 and use a marker to draw a 0 on ball 1 to make it say 10. Step 2, add balls 11-19 and use the marker to draw a 0 on ball 2 to make it say 20. Step 3, add 21-29 and change the 3 to 30, and so on. How many balls are there at the end? Well, if you claim that ball n is in the vase at the end, I will respond that ball n had a 0 drawn on it at step n, so that ball is no longer labeled n. So, we never removed any balls yet it can be argued that the vase ends up empty.

Like the exception for zero. When you multiply 2 polar form complex numbers you multiply the magnitude and add the angles. That's true for any number, except zero. Zero doesn't have an angle so you have to make an exception and just ignore the angle. It works because the answer is zero which doesn't have an angle. Sometimes infinity is defined as a number divided by zero, eg. 1/0. If we say that infinity can be defined then, for the same reason as above (because with polar form division you minus the angle), infinity would have to have no angle. I see a lot of similarities between zero and infinity.

This is one of those beautiful cases where the limit and actual infinity diverge. cause in the limit, as we approach infinity, we are building up infinite balls, yet if we do this process for every number on the number line, we end up with no balls at all.

Okay...sooooooooo... first of all, let me get realistic factors out of the way. Neither time nor space are infinitely divisible: there are Planck lengths and Planck instants, and those are pretty much how our mathematical probabilistic quanta work. Light has a 100% chance of moving onto another Planck voxel in the next Planck instant, and anything going at a lesser speed has a lower chance of doing that. (and dang I just realised that if you compressed yourself too much by going too closely to the speed of light, space itself would push you apart as if you were pushed against a wall by a very heavy force. Either that, or Planck lengths would compress, but then what would be the point? We have special relativity for god's sake.) That means you can't do this in real life, because eventually you'd stop. So we have to do it mathematically. Doing this, is like, doing it every minute for eternity. Now, it's kinda not possible to reach an infinite value using only finite numbers, unless you have infinity of them already - in which case, you pretty much have infinite numbers. That means, to finish doing this, you have to get to the end of eternity. Sounds crazy, right? Imagine eternity having an end - that ruins the whole concept of eternity! So I'm going to explain what infinity even actually is. I kinda wrote it somewhere on a video about infinitesimals but I stopped so I'm gonna have to get back to it again. Okay, so, real numbers (boi don't get me started by complex numbers, but I assume they'd start with rectangles) can be defined as a line segment with one end on 0 and the other end at where their value is. Obviously this has a bunch of applications, you can shift the points to get differential values or just measure the length of them to get absolute values, but basically, that's how it works. Positivity is a direction, right? (By the way, that's exactly why infinitesimals are not basically zero - mostly because they're positive.) Well, infinity just basically has a point at zero, and a direction leading toward high positive numbers. It doesn't have the other point. It's a line semisegment and it's pretty much the difference between a vector and a scalar (yeah I didn't get too far into that cram book about physics). So if you were to shift the points one up, would you be adding one because all of the line would be shifted one up, or subtracting one because you just moved the line one up without doing anything to infinity? Neither. Like I said, you can't reach an infinite value using only finite numbers, and I'll be darned if you can do anything to infinity using only finite numbers. I mean, you could use a negative hyperoperation and use an infinite number in it (the result of ∞÷∞ is the set of all numbers (every single one) between an infinitesimal and infinity, and the result of ∞-∞ is pretty much the set of all real and hyperreal numbers between negative infinity and infinity), but I cheated because I used infinity somewhere in both entries (although it does raise the question of what the heck an infinite hyperoperation would be like). If you were to do it mathematically in a finite fashion, then, well, you'd never end. Having an infinite series pretty much is a cheap excuse to get to the bottom of it, but since you can do that in mathematics, let's try. Okay, so, uh...when Jade does it, she ends up with a set of integers multiplied by 10, and when Blade does it, uh, well, she has an infinite supply of balls, so since ∞+∞=∞ then ∞-∞ will contain infinity as well. So what she'll do, is, she'll...don't you see? If she REALLY has infinite balls, then somewhere in an inaccessible pseudospace, she'll have an infinite supply of balls marked with the simple lemniscate as well. So do you know what she'll have if she tries? Assuming she does in fact get to the end of the stock, which incidentally doesn't exist, she'll have a set of integers, and pretty much drop off at infinity. She'll have infinitely many balls she didn't use and infinitely many balls marked ∞ she did use. I mean, since ∞×∞=∞ as well, the most basic infinities are really nothing compared with true infinities (although I only believe in different types of infinities because of divisions like countable and uncountable infinity (and that was after the explanation)). Honestly, uh, well, I'm still assuming they're humans, so let's replace them with mathematical variables with their names. Well, uh, yeah, Jade would still have that set, but what about Blade? I mean, I know she eventually takes away every number from the jar, but I'm still thinking finite here. She puts 9 times as many numbers in the jar EVERY TIME she makes that happen. I think we can all agree that ∞+1=∞. See? No difference. Although, some mathematicians don't - like they think that, uh, what's the symbol? **checks **ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-TCy6r3XOzqc.html** They think that ω is different from ω+1. Personally, I think they're all lunatics for thinking you can just add one to infinity and make it larger than it already is. I mean, since ∞×∞=∞, and you can pretty much match an infinite set of any division of finite numbers to a set of integer or even infinitesimal numbers, infinity contains itself, STILL. (Oh and they also think there are nigh-infinite numbers, but I think they're -even bigger- lunatics. I think they should just put their math into making Touhou bullet patterns - then we can have a lot more interesting patterns than just flowers and spark patterns. I've seen those random mathematical art simulation projects on RU-vid and Scratch.) So I think the idea that ω is any different from ω+1 is preposterous. I mean, I spent so long I forgot what point I was going to type here other than the one people already disagree with, but like...come on. Just...WHY??? Like it's so insane, think about it. Infinity doesn't end. Assuming Blade reaches it (and oh yeah, that's the variable in the scenario), she'll still have an infinite number of balls she didn't use because if she used them all, well, that's just silly. Uh, so, ｏｋａｙ　ｗｈｅｒｅ　ｔｈｅ　ｈｅｃｋ　ｗａｓ　Ｉ okay so let me type this again _Blade_ repeatedly puts 10 balls into the set _jar_ and takes one ball out every time infinitely. And since it's an infinite series, she's forced to stop. That's one way I can prove that she has infinite infinity balls she didn't use. I mean, if the set ended, that...uh...yeah that would, uh, pretty much defeat the point of infinity. Damn it I'm still having trouble visualising Okay, the argument. At the end, she eventually returns every ball. Even though she put another 9 in there every time. I still think you're thinking finitely...? I mean, when she gets to infinity, she'll have 9 times the infinite balls in there. And, like, 9×∞=∞. Uh, so, once she gets to that pseudospace, I think there'll be a big hole of balls that went to _jar_ with no exact point of starting. So it's very blurry. no actually scratch it. it's a blur with no boundaries whatsoever. We're already in the pseudospace and this is infinity we're talking about. So there'll be an infinite set starting at 1 with the balls she returned, and an infinite set starting at infinity with the balls she left in the jar. There. That's my answer. Bye now. (also of course Blade would think there's nothing in the jar - it has infinite hammerspace, so she wouldn't even be able to hear the infinite balls of infinity tumbling down as a result of her tipping it down lol)

The fallacy is that the definition of comparable states is that you wait until you have compared all the elements in the sets. when you finish, While with the infinite sets you ignore this requirement by trying to predict what the outcome will look like and accepting that answer if it sounds logical without providing proof of the logic that lead to your answer when you have finished grouping them, but with a infinite set, you can never finish grouping them so you never come to your decision point you are basing your answer on. To put it more, simply you never meet all the requirements of your comparison method because you always have more cups of chocolate to hand out and more cows wanting chocolate.

If at every time step you are adding ten balls and removing one ball, it is not possible to have less than infinite balls in the vase. It does not matter if you are removing balls 10,20,30... or balls 1,2,3... You are always adding more than you remove, assuming in both cases you start will infinite balls outside of the vase.

division and individuality are arbitrary ideas, which is a way of saying that math is just an arbitrary idea like a language. languages aren't absolute but they can still convey meaningful and true ideas but they fall short in many cases as well, so does math.

If you look at velocity, most people agree we can't move matter upto the speed of light, as the amount of energy to move a massive object increases to infinity at C. And a massless photon has a speed limit (Ignoring the expansion of the universe). So perhaps in the mathematical language infinity exists, but one day we'll discover, with maths, that in reality theres a limit.

Really creative way of making this paradox accessible. Loved it!

There is a finite limit we measure time with. The Planck limit. Can't be divided again. We get 9 in the case for step until we divide 2 Planck by half

Correct me if I'm wrong, but if we use Ordinal numbers instead of Cardinal numbers, wouldn't it show that both vases will have an infinite amount of balls? * If one uses Cardinal numbers, it isn't possible to assign identity to any balls because all Natural numbers have been used up, hence the reasoning "there are no balls left => empty" * If one uses Ordinal numbers, it is in fact possible to assign identity to balls even if you remove an infinite amount of them, and hence it is possible for 'Blade' to still have balls left in her vase. I draw inspiration from Vsauce's "How To Count Past Infinity", where it is presented that if one uses a symbol for the order that each ball was put in, one could count beyond infinity by labeling the Ordinal numbers beyond with ω+1, ω+2, ... and so on. Therefore it seems possible to reason that one can remove the first ω balls, and still have balls ordered with ω+1, ω+2, ... and so on.

If time is infinitely divisible then you will never get to "noon" and therefore never be able to check if you have an infinite number of balls or none.

If you assume the time can be divided infinitely small you never reach noon. The number of ball would approach infinite.

But in blades example, the vase should not be empty. Because with her example she will never get to noon. She will always be forever adding balls and removing one, just doing it faster and faster. But one of the infinite will not ever equal out to the other. Yes it is a 1:1 but it's set at a staggered starting point. So the correct awnser is that there will always be balls in the vase, yes for any time stamp there will always be one where you removed that ball, but for every timestamp there is there is also so many balls left not yet removed. So there technically should be a 1:1 btwn the 2 examples showing that at any given step of timestamps there is the same amount of balls left in the container. (X+10)-1 where x = the chosen timestamp. 1st timestamp, 2nd etc

i have a different solution i might be wrong but please tell me where i wrong at so if i take out ball 1 and then 2 and 3 ... it should still be the same as i take out ball 10 and then 20 and then 30 cause {1,2,3,4,5...} is equal to {10,20,30,40,...}(both are infinity) so both vase should be empty right ? (sorry for my bad english)

Always go back to the limit of the series. The individual terms of a conditionally convergent series can be made to sum to anything you want if you change the order of the terms, *and hence the series*.

I guess I should add that infinity is more of a journey than a destination, where the only way that the vase could be empty is if you quit adding balls and use the rest of the steps to empty it out. In terms of time, time has no choice but to be infinite, but in order for time to be useful to us, we need a reference and our earliest reference is the moment of the Big Bang. There is certainly a "before" the Big Bang but we have no references to this time. Just because time is useless to us doesn't mean it doesn't exist in the same way we have energy that is useless to us exists. Time is not a thing, it is a measurement, which is why the reference is necessary for it to have meaning for us.

I always had a problem with this mapping. My objection is as follows. It is clear that the numbers 1 to 9, 11 to 19 etcetera are not in the set that contains numbers that are multiples of ten and that every number in the set that contains multiples of ten is in the set of positive integers. So to me it is clear that there are more members in the positive integer set. So I would say that the set of positive integers is more "dense".

What if in the first example, where taking out ball 10 and 20 and so on (where you would be left with infinity inside the vase) you changed the number to 1 and 2, and so on, while changing the number of the original 1 and 2 to 10 and 20 and so on. In effect, in reality you didn’t take out any different balls than what left you with infinity, you altered the numbers. But in so doing, you would then be left with zero.

This reminds me of the reimann series theorem that states for any real conditionally convergent series, there exists some permuation in which the series can converge to any given real value, and there also exists a permutation in which the series diverges to infinity and another where it diverges to negative infinity. Which brings up another weird fact. This means that for real conditionally convergent series, the communative property of addition does not apply. Though I dont know if it can be applied to this specific problem. If we use the number of balls, clearly it diverges by the divergence test. Maybe you can go by percent of balls or something?

What is the name of the song at the end of this video?

One of the original assumptions (infinitely many steps are performed) requires that "noon" can never arrive. Asking what happens "at noon" is therefore meaningless - the paradox (two contradictory answers) arises because there is no "correct" answer. Unless you start with different initial assumptions, there is no "noon" to ask questions about. All you have are two numbers, both of which are approaching infinity.

This channel deserves more view

"I'm on a budget." 🤣🤣🤣

What if you're looking at an infinity that goes both ways compared to an infinity that goes only one way. For example: an infinity of all the negative numbers plus all of the positive numbers (...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...) paired with an infinity of only the positive numbers (1, 2, 3, 4, 5...). How do you create a 1::1 pair when there is no real "first number" in the first set? It seems like no matter where you start the pairing, the first set is still going to have an infinity of numbers further into the negative that will never be paired.

Shouldn't the equation be written X = (Y-n)? using the example of 1 to infinity being X and 101 to infinity being Y, the equation would be X = (Y -100). You are discussing is Zeno's paradox where there is no end to the division of by half. In your example, you use a time period. That time period comes to an end at 12:00 Noon. Wherein Zeno's paradox the time period would not end but go on for infinity. You left the paradox and entered into the real world. To enter the real world you need to subtract a number, no matter how small to make the jump from the paradox to the real world. Please correct my assumption if it is incorrect.

That's why you can prove 1 = 2 when dealing with infinities. A circle doesn't exist in the real world, because it would require infinite points. But, the concept of a circle allowed us to discover or shall I say invent a lot of new Math that is used widely throughout Physics.

Number is on the vase, rotate 8 90 degrees, at 1st thought. Dividing a timestep near a plancksecond by 2 poses a problem. Time is quantized in planckseconds. Nothing can happen before the next plancksecond. timelimit=plancksecond. But how fast can I put 10 balls in and take one out? timelimit=a seconds. After every step remaining time = 60/(2^(number/9)) = timelimit => ln(60)-ln(2^(number/9)) = ln(timelimit) = ln(60)-ln(2)*(number/9) => ln(2)*number/9 = ln(60)-ln(timelimit) => number=(ln(60)-ln(timelimit))/ln(2) rounded down to integer *9.

Surely there's always a ball "lag" though in blade's example even though there's always a step it get's taken out there's always 9 more balls getting put in at that step too which means there's always a next step and no way of ever reaching the point where you take out the last ball of the vase so it's never empty and if you iterate long enough the 9:1 ratio goes to infinity?

I might have missed it but how do you ever get to noon if you keep dividing the time by half as you approach that time?

You forgot about quantization. In the thought experiment, neither Jade nor Blade accounted for the fact of observance that for material composition to be, we have regulated and set distinct levels of quanta in the same way how frequency of impact can only be transferred, made out, and received in distinct measures; anything beyond and it's like continuous pushing on a swing and is a state whereby the local observance really no longer sees the swinging and this in-itself and its sustain forms material composition in an elseworld's relative scale of observance, when you have system ran stable enough such that you can have the swing pusher, keep it fed, keep the swing maintained while swinging, and so forth, then this is evidence of universal preservation and shrinkage (think DC Supergirl when Brainiac of that one world shrunk his Earth in a canister to be revived later after experiencing the aftermaths of Crisis on Infinite Earths.) And so what can be said in simple terms is, the ONLY set of balls that will ever MATTER, is the set that is within your renderable state of quanta (meaning the smallest tick size of a second tick hand you can create within the confines of the operations of a clock for a clock to be even recognized and agreed upon as something functional, still observed to be counted, and to do so in good health without your own collapse). And so once again... we go back to the age old quotes by "" of how the mass majority should think deeply but is very much insane... Sane is when the thought trailing will not forget to hold itself admitted to the confines (hospitality) of the host order - that is say your average man's second tick identifies on a scale down to 60 ticks per minute, with some watches running perhaps 360 ticks per minute, and so forth. This then is your measuring apparatus, your significant figures, and is to be the written test answer to the question of how many balls are in the jar (assuming you can afford to buy that many balls and the jar is fitting too of the balls, and so forth). In other words: Both Jade and Blade are wrong due to incompleteness of brain formation and there was no paradox. Also as a side note, to the approaching of the various infinities like is an infinity that doesn't include the first hundred = to one that does, and to that I'll tell you, you forgot to account for the fact that your diversifying of case itself was the reason and case being for what is missing. In other words, the infinities are equivalent with you the local observance holding account of it factored in as an integrated part of system entropy, the fact that we'd have to break your thought processing down and disseminate your entirety as a fill for the 100 thus giving us still 1+1=2, rather than just assuming pairs, which if you will remember back to grade school is a making of an ass out of you n me. -Dog harsh and strict sounding maybe, hope you didn't get offended. Not really directed towards you, just the answering

Revamped version of ‘you can never go from a to b, because each time, you have to get half the way there’. And I the argument discards the fact that at any time t, you have 10t - t balls. At any time t, you never have a 1:1 correspondence

There's a limit for dividing time with the Planck time. Therfore the amount of balls is finite. Altough very large. I'm sadly way too lazy to do the math and it would likely break my calculator.

Everyone has one of those days. Forgot how to count, surrounded by cows, they all want milkshakes. Thank you for helping us with our everyday problems!

Most of these "paradoxes" involving infinity make the same mistake: They treat infinity as if it's a number.

This seems like it should make sense, but I dont think it works for a random drawing of balls. Because in a random draw, I could pull out only even numbers, leaving my vase with only odd numbered balls. Does randomization affect the correspondence and at which point does it?

Infinity is not a number and cannot be treated as such is basically the long and short of it. It's a property of a set describing it's size, it's more akin to a colour in what you can do with it mathmatically, ie very little.

2:35 “I thought you said to remove ball 1 at the fist time step and ball 2 at the second time step and…” No. She said to remove 10. Then 20. Then 30. Etc.

Seems like the 'infinity paradox' is a side-effect of connecting physics and math without an adequate symbol to represent the concept of 'unknowable', or 'indeterminate' on the physics side before handing information off to the math side. The concepts of no-count (zero) and unbounded-many-count (infinity) are interesting and useful, but applying them to certain physicsmath calculation results in dithering paradox. Counting and measuring things in the universe - enumeration - is almost entirely physics except that counting relies on concepts about numbers and sets (i.e. math) to derive meaning. Performing operations on numbers and sets is purely mathematical - except everywhere the intent is to calculate something meaningful or useful related to the universe (hehehe).Anyway, similar to zero and infinity, it seems like an 'unknowable' symbol might help to identify indeterminate inputs in order to resolve math operations that produce dithering paradox instead of invalidation, or 'unknowable' output ranges - like divide-by-zero.