Hey, Vsauce. Michael here. Now, one of my favorite

treats of the holiday season is Gabriel’s cake. It’s a super solid

based on Gabriel’s Horn that you can make right at home, as long as your home

is infinitely large. Okay, all right. Now, the first thing you want to do is bake a

cake. I prepared this cake earlier. It’s a real beautiful cake. It’s a little large,

but I bet you I could eat that whole thing in a day if I tried. Am I right?

Okay, now the second step is cut the cake in half. Notice that while I cut this cake in half you don’t add or create any new cake but

the surface area of the cake has increased. It used to be completely

covered and now we’ve got two regions on the inside that aren’t covered. The next

step is to cut this half, one of the halves, in half. Here we go. Again, the volume of cake on

the table is the same as it was in the beginning but the surface area is going

up. As you may be able to guess, the next step is going to be to cut one of these

quarters of the original cake in half. It’s getting pretty thin, but all you have to

do is keep this up – cutting halves in half and half and half and half forever and

once you’ve done that, well, you’re almost there. When you finish cutting, stack the

halves on top of one another in order, like this, to create a beautiful tiered

dessert. Because you have an infinite number of thinner and thinner slices

when you’re done stacking them the cake’s vertical height will be endless.

Such a cake has an interesting mathematical property. Its volume, the

amount of cake it contains, is clearly no different than the amount you started

with, but its surface area is infinite. It’s a cake you can eat but not frost.

You would need an infinite amount of frosting to cover the whole thing with a uniform

coat. An object with finite volume but infinite surface area doesn’t need to

be endlessly tall, by the way. There are bounded super solids, like this cube,

with an infinite number of smaller and smaller circular holes.

Of course, building these objects in the real world comes with some obvious difficulties.

One is the fact that the amount of steps required to complete the construction of

Gabriel’s cake or any other super solid is literally infinitely many.

And infinity isn’t a number that, you know, you eventually get to.

It means unending. There will always be a next step, another piece to slice in half again.

By definition, an infinite sequence of tasks has no last task. So you could never

finish making a super solid. Or could you? Enter the Supertask. What if instead of taking the same

amount of time to complete each step we accelerated as we worked and did each

step in half the time as the last? For example, let’s say you wanna make

Gabriel’s cake in just two minutes. That’s easy. First, cut the original

cake in half and then wait a minute before making the second cut. Wait half a minute before making the third, a quarter of a minute before the 4th and so on.

Always waiting for half of what’s left to pass before cutting again. Since you can keep

dividing in half forever, there will always be another step. Infinite actions

within a finite amount of time is a supertask. The strange thing, of course,

is that while supertasking no matter how many steps you’ve already completed

there will always be an infinite amount of steps ahead. But yet, when time’s up,

you will have finished all of them. What we’ve just described is similar to Zeno’s

famous paradox of the dichotomy in which the Greek hero Achilles runs a short race

that he can clearly finish, I mean, people finish races all the time,

but how exactly it’s finished is a mystery. Because first Achilles must

cover half of the race’s distance and then half of what’s left and half again

and again and again. Since there will always be another half way point to

reach, the number of subdivisions infinite, there is no final destination,

after which Achilles’ next stop is the finish line. But yet Achilles obviously can finish.

He somehow reaches them all despite the fact that during his journey he always

has an infinite number of steps left to reach. Think about it this way.

What if at each half way step along the way Achilles is required to hold up a flag.

At even steps a blue flag and at odd steps a red flag. Blue, red, blue, red,

blue, red, the flags will alternate faster and faster. But when Achilles finishes

the race, which flag will he be holding up? To be holding up either flag would

seem to suggest that a largest number not only exists, but is either even or

odd. Now, to this it is often suggested that the real problem here is that Achilles

isn’t actually doing infinitely many things.

He’s just taking a few strides or whatever. A true supertask requires doing

infinitely many distinct actions. Fair enough. But even if Achilles ran the

race in a staccato fashion, stopping at each half way point for half as long as he waited

to the last one, the scenario would still be sensical and logical and would

complete in a finite amount of time. It has even been shown how a staccato

runner could move to avoid discontinuous velocities and accelerations.

The point is, fine, infinite distinct actions are at least

logically possible to do in a finite amount of time, but only logically.

Only as a mathematical abstraction. Supertasks are obviously just products of our

imagination. Because in the real world there must be some smallest amount of

space and/or time that cannot be meaningfully divided in half again.

Once you’re within one of these distances of the finish line, the only possible next position you can have is the finish line. That certainly seems fair.

After all, it’s believed that in reality there is a smallest meaningful distance – the Planck length. Any interactions between or observations of particles

more accurate than this span make no sense within physics as we understand it

today. And the time it would take to travel a Planck length at the fastest

speed possible, the speed of light, is the Planck time. So these seem to be the

briefest pieces of space and time any known force in the universe could act

across. Whether that force is gravity or a force from Achilles’ leg muscles.

But does that resolve the crisis or is it just a way of avoiding the problem

altogether by saying “it doesn’t matter, because it won’t be on the test?”

As John Earman has complained, it seems to me unattractive to make the truth of

mathematical statements depend on the contingencies of space-time structure.

Maybe Zeno’s dichotomy tells us less about motion and time and space and truth

than it does about our impressive abilities to confuse ourselves.

And that’s what makes supertasks important. They’re a contact point in uneasy

handshake of sorts between the universe we live in and the brains inside us.

Let’s play around with some. They’re not all the same. Some converge, like Zeno’s dichotomy,

others converge but in ways you may not expect and still others refuse to

cooperate in any way. They diverge. A great example of a supertask whose

behavior diverges is Thomson’s lamp, a famous supertask devised by James F. Thomson. Imagine a lamp that can be turned on and

off as quickly as you desire. What would happen if you turned such a lamp on and

off Zenoianly? Well, let’s find out. Just set a timer and turn the lamp on. Then wait

one minute and turn it off. After half a minute turn it back on and then off

again after a quarter of a minute, on again after an eighth of a minute and so on. Waiting half as long to flip the switch

each time as you previously did. Now, as the number of switching grows without bound,

the total time elapsed approaches just two minutes. The lamp will be turned on

and off an infinite number of times in just two minutes.

So after two minutes, will the lamp be on or off? Well, by the definition of infinity,

there is no last step. There is never a switching not followed by another.

So the lamp can’t be on, because whenever it’s turned on, it is immediately turned off

in the next step and it can’t be off because whenever it’s turned off it’s

turned on right afterwards. What’s the answer? It’s easy to say something about

a supertask when its partial sums converge, but when they just oscillate back

and forth forever… Hmmmm… Zeno objects do this. Imagine building a meter

high cube. As with all supertasks, you construct at an accelerated pace.

First, you put down a green half metre tall slab, then a quarter metre tall slab that’s

orange, then a green eight metre slab, then an orange 16th meter slab and so on,

until there are infinitely many layers of alternating color. Now, when you look at the cube from above, what color will you see? Orange, green? Well, it can’t be orange

because every orange layer is covered by a green one. And it can’t be green

because every green layer is blocked by an orange one above it. What if we had a

machine display each digit of Pi in order at a supertask pace? After its

finite runtime, what would be on the screen?

The last digit of Pi? Well, that’s impossible, right? But how could it be anything else? A supertask allows us to exhaust an infinite sequence. Paul Benacerraf delivered what is often considered the

best response to these confusions. Is Thomson’s lamp on or off? Is the cube orange or green? The answer

is we don’t know, because these questions are incomplete. Thomson’s lamp could be

on or off or broken. The cube could appear orange or green or something else.

But the supertasks, as stated, don’t let us figure out which. I may as well ask

you if a lamp hidden in a locked room is on or off. It’s definitely one or the other,

but I haven’t given you enough information to do anything but guess. Supertasks like

these describe an endless sequence of tasks and then ask us about the end.

But we can’t determine an outcome because, although there may be an end to their

duration, there is no end, no final member of their actions.

They must be reworded or coupled with extra assumptions in order to be solved.

For example, if we assume that the switch used on Thomson’s lamp can only be all

the way on or all the way off, we can’t determine where it is after the supertask.

But if the switch is, say, a bouncing ball that completes the circuit

turning the lamp on each time it bounces on a metal plate, an outcome can be

determined. If the physics here are ideal and the ball bounces half as high and

half the time as it did on the previous balance, its sequence of

bounce heights will turn the lamp on and off an infinite number of times in a

finite amount of time. Although this bouncing ball has no penultimate state,

no second-to-last bounce, it does have an ultimate state, a final one, resting on the plate. The circuit will be

complete and the lamp will be on. You can also describe a switch on Thomson’s lamp that leaves it off. Sometimes, the next state after infinitely many isn’t

paradoxical because of lack of information, but because of a surprising,

or non-intuitive, discontinuity that occurs there. The Ross–Littlewood paradox

is one of the greatest examples. Imagine a giant urn that can hold an unlimited

number of balls. Now, imagine that you have an unlimited supply of balls, each

with a unique natural number written on it. All natural numbers, in fact, since

there’s no end to how many balls you have. Now, working at an accelerated Zenoian pace you move the balls to the urn 10 at a time, but in a weird way. At step one, you place

balls number 1 to 10 in the urn, but remove number 1. At step two, a minute

later, you place balls 11 to 20 in the urn and remove ball number 2. At step three,

you place balls 21 to 30 in the urn and remove ball 3, and so on. Upon the completion of the supertask

how many balls will be in the urn? At first the answer seems obvious. At each

step you are adding 10 balls and subtracting 1, so a net of nine balls

is added each time. 9 + 9 + 9 + 9 forever, the series grows without end. Infinite nine’s means infinite balls at

the end. But here’s the problem. At each step, the ball with that step’s number

written on it is removed. Ball 1 is removed at step one, ball 2 was removed

at step two, ball 12-googol is removed at step twelve-googol. Since there are an

endless number of steps for any ball number, there is a step number at which

it is removed. So although the urn’s ball population grows without bound during

the task, after the supertask the number drops to 0. It gets weirder. Here’s a second, seemingly identical

method. Instead of beginning with balls 1 to 10 and then removing 1, begin with

balls 1 to 9. Then write zero after the “1” on ball 1. For step two, add balls 11 to 19 and draw a zero on ball 2, making it say 20. For every finite step, both methods

results in identical earned contents, but after infinitely many steps, the first

leaves us with no balls and the second leaves us with infinitely many balls

written on which are all the natural numbers, each followed by an infinite

string of zeros. Both are discontinuous at infinity, but dang, in very different ways.

The bigger question now becomes, “so what? Who cares?” You will never have an

infinite number of balls and you will never have a large enough to urn to

hold all of them. You will never build a lamp that can turn on and off arbitrarily fast. We cannot investigate

time or space past a certain smallness, except when pretending, so what are supertasks, but recreational fictions, entertaining riddles? We can ask more

questions than we can answer, so what? Well, here’s what. Neanderthals. Neanderthals and humans, us,

Homo sapiens, lived together in Europe for at least five thousand years.

Neanderthals were strong and clever, they may have even intentionally buried their

dead, but for hundreds of thousands of years, Neanderthals barely went anywhere. They pretty much just explored

and spread until they reached water or some other obstacle and then stopped.

Homo sapiens, on the other hand, didn’t do that. They did things that make no sense

crossing terrain and water without knowing what lay ahead. Svante Pääbo has

worked on the Neanderthal genome at the Max Planck Institute for Evolutionary Anthropology

and he points out that technology alone didn’t allow humans to go to

Madagascar, to Australia. Neanderthals built boats too. Instead, he says, there’s

“some madness there. How many people must have sailed out and vanished on the

Pacific before you found Easter Island? I mean, it’s ridiculous. And why do you do

that? Is it for the glory? For immortality? For curiosity? And now we go to Mars. We

never stop.” It’s ridiculous, foolish, maybe? But it was the Neanderthals who went

extinct, not the humans. Maybe it’s only a fool who will perilously journey out to

what might not be there. And maybe it’s only a fool who will ask about supertasks,

about infinity. But if you want to solve problems, you don’t just solve the

ones that are there, you find more and make more and go after the impossible

ones; fostering a love and obsession with problems is how you solve problems. Antoine de Saint-Exupéry wasn’t a

mathematician, but his advice fits nicely here. If you want to build a

ship, don’t drum up people to collect wood and don’t assign them tasks and work,

but rather teach them to long for the endless immensity of the sea. And as always, thanks for watching. Supertasks are cool, but super gifts are

even cooler. That’s why I’m excited to announce this year’s Vsauce holiday

box. This thing comes loaded with exclusive Vsauce stuff and science gear,

plus all Vsauce proceeds go directly to Alzheimer’s research. I’m really proud of

this box. You can pick one up at geekfuel.com/Vsauce,

link down in the description. There’s a limited amount available, so

don’t hesitate. And as always, thanks for watching.

We’ve got a 50/50.

damn imagine when we create a computer that can turn on and off a lamp after half the time passed etc. and we find out what would happen after two minutes

Plonk!

Michael: The lamp can either be on or off. What state will it be in?

Me: It'll be broken, I guess…

The cube will have a color of green mixed with orange, because the layers will be so thin that they become translucent. Boom. Problem solved. 😀

It annoys me that he didn't cut the cake with the other line that bisected it

0:57 just staring at the knife

12:36 it's not definitely on or off, it could be schrodinger's lamp

I like how as soon as Micheal picks up the knife the laughtracks ended.

Such madman.

This the meme i was wanting to find!!!

(7:55) Let's look up these units:

a light-year is a distance the light travels in a year: 9.46×10¹⁵ m or 9.46 Pm (petameter)

a light-second is the distance it travels in a second: 3.00×10⁹ m or 3.00 Gm (gigameter)

a plank time is the time light travels in a plank length: 5.39106×10⁻⁴⁴ seconds

then a meter time is the time it travels a meter: 3.33564×10⁻⁹ seconds

That world is heaven for michael because every pun he make somebody laugh

This "problem" is solved. There not an infinite number of dots on a circe because that what makes an circle is not infinite. At one point you will reach a planck distance between the points and there is nothing in between. There is not even "nothing" in between. That space or that value simply does not exist. You can not ignore the quantum state of things. When you are moving you are actually allways moving in planck distance one at the time. You are moving digitaly and not analog. You are moving from one state to another. For you it seems like you can move infinitly long period of time and changing infinite number of states from one point to another. But the smallest divider is the planck lenght. At one point you will be one planck lenght away from the line and your next state will be the final one. On large scale your state changes from state start to state goal but in between you are actually changing a certain number of planck lenghts and planck time. Infinity is a theoretical term. Planck number is the limitation, a frequency for the universe to change from one state to another at once. Like your pc chip changes from one state to another in lets say 3GHz, the universe has its clock and its planck frequency. It is digital not analog. You cant divide infinitly because at one point there is nothing to divide, no space, no matter, no time…actually there is not even "nothing". Lets explore more. Lets say you have a solid stick made of wood. It is long 10 million light years and reaches some other planet in universe. If you push the stick will you be able to poke that other planet with that stick instantly? Well no. On that scale you will notice how you impuls travels across the stick. Changing the state of each molecule in it from one state to another, to one point in space to another. Even on this scale you still would not notice the matter moving in planck lenght or time. The number is just too small.

(10:18) the lamp will be broken.

you will never have an infinite number of balls. Thanks, michael.

legends say he's still taking sweaters off

0:58 When you’re doing a baking TV show and contemplate killing everyone in the audience.

mental masturbation. hubris. misidentification posing as a problem.

In order for the cake to be physically sliced at greater and greater frequencies, I believe that at a certain point the speed of the utensil being used would reach light speed. With our understanding of light speed being the fastest possible velocity, this would define a definite limitation to these problems. From what my brain can gather, any physical representation of a super task would have this limitation: eventually requiring a speed that is theoretically impossible. Or is it?

Wouldn’t you get to the point where you would have a single line of atoms if you cut it you would split an atom and die

my answer to the cube problem: there will be a completely different color on top. the reason why is because if the layers are getting thinner and thinner, eventually they'll get so thin that they'll become partially see-through. because of this, the colors will blend, and there will be a new color on top.

With an accelerating I write 10 comments and I remove one while each step I like and the next one I dislike the video.

Anyone know the music he used for 3:38 I can't find it and it's stuck in my head

Vsauce is filmed in front of a live studio audience.

0:12 “You’re breathtaking!”

Infinant

You can not keep slicing cake in half. Sooner or later you're going to be at the atomic level where it will be impossible to half it.

This is some Green Baby shit.

Also, this video is stupid. "Here is a task without an end… What happens at the end?". Nothing you moron, you just said it didn't have an end.

The lamp is dim

Make cake

1:28 Steve

is hungrytask manager has stopped responding. Restart?Wait…

So Michael ISNT Vsauce?

He says: Hey Vsauce, Michael here.

We are Vsauce.

Synchronicity proven! I'm watching Welt Am Draht ("world on a wire" with English subs) and the dead guy draws Zeno's dichotomy before he dies. I've never even heard of Zeno before yesterday (yesterday because I keep turning the movie on when I'm dead tired).

With the balls one, what about putting 10 balls in and taking the 10th out, you would have an infinite amount of balls in the urn, and an infinite amount of balls out of it.

Now i'm hungry

the last color on top of the cube would be brown.

Where is Schrödinger when you need him. Probably messing around with that damn cat again.

The cube would be grorange

In the beginning he said it can be done if your home is infinitly large, however if the volume stays the same as the cake shown then surface area doesn't equal 3d space. So therefore you wouldn't need an infinitely large home

The answer is A: All of the above.

If you make pi a length, 3,1415….. cm. Is it possible to make something that long?

My birthday’s coming up,

WHERE’S GABRIEL’S CAKE??Neandertalls

for more chaos do 25% each time or 5%

Youtube acting weired. No of replies are 13, when I open reply section there are 20 replies. Whats happening😨😬😱

14:40

this has such a too many cooks vibe I'm waiting for the machete dude to pop out

One interesting thought is the point of reference. If Achilles was aiming to reach a spot double the length of the end of the race, he could reach it just fine. But the thing is, there is no real such point of reference, or a point that one is infinitely aiming to reach. In all reality, nothing really depends on that goal, but yet more on what ends up actually happening. For example, if the audience thinks that Achilles is aiming for double the length of the race, he makes it just fine. But if someone else thinks he is aiming to reach directly at the end, he doesn't. The point is, constructing an absolute reference point is impossible, and unaffected by how we think. While mathematically, we can create our own reference points such as shown in Gabriel's horn, this is simply not the case in regular movement.

I have always been at odds with peole wising for immortality, but i really dont think those people grasp the concept.

21:04 lol

He keeps cutting the cake in half…it must be homeless. And in 2025.

I'd argue you need an infinite number of balls to sail unexplored open ocean.

I'd argue you need an infinite number of balls to sail unexplored open ocean.

now i fukking want cake!!!

2. I bet you could divide Planck in half . . time and space . . maybe the guy too

Mike stop the cheesy videos. Too much cholesterol.

So uh… You're going to eat that cake???

If you were to keep cutting the cake in halves you would eventually turn it into an atom bomb.

-Kim Jong-un wants to know your location.

eventually the cake is on the molecular level and once you cut it in half there it isn't a cake anymore

I counted, he said the word “balls”

30 timesin this videoHmm… Supertasks and supersolids… Clever…

New sweaters at:

3:40

9:15

10:58

14:10

17:08

Why didnt he take then off in a super task method, each half way point?

I'm no expert, but I would like to believe that the end results of these supertasks would be a "superposition" like in quantum computing. The data points occupy two positions at once.

You just made me want to eat a cake

Wait what? He only has 14 Million subscribers?

Why

Vsauce never told anything… But asked every time. Dnt have answer for any thing..

10:43…after so many attempts lamp will burn off…and finally fuck off😉

This is possible if it is always possible to double the processor of your computer in half the time you did last time. I guess if there was a cookie like black mirror making the processor would be in for a treat

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but hey thats just a theory a game theory thanks for watching

well youtube may have infinate time but you do not honey is a browser that helps you find free coupon codes instantly… jk honey doesnt have infinatly fast CPU like youtube

Why din ur shirt change faster and faster forever

Well now I'm hungry.

5:50 i expected him to take off

But you get to the point where the cake is 1 atom thick, then what? It's not infinite. Actually, it's far from it.

The lamp burns out.

Achilles gets confused about which flag it was.

My mind just broke

In calculus we call this a convergent series

Is infinity a paradox?

anyone else just wanted him to eat a piece of the cake?

I told you. The cake is a lie!

Our primitive and bounded minds are unable to comprehend the complexities of infinity, however, a more complicated being would be able to solve for and think of this as the equivalent of their “10th grade” number theory, geometry, algebra, analysis or whatever. Let that sink in.

You will never have an infinite number of balls

That’s what he said

The more you know

"Frosting a Cake" Like if your mind is in the gutter.

Dang, you always find a way to put some nonsense about evolution in your videos don't you. Neanderthals ARE humans from not that long ago.

Wat

Achilles arms must hurt after that race.

we don’t know what flag he’ll be holding because he’d helicopter

"It's a cake you can eat… but not frost." Fucking brilliant. And now I am dead.

@Vsauce You know what else is a supertask? My homework.

But if it takes light, the fastest thing ever, the shortest amount of time existent to travel the shortest distance existent, then how far do slower things travel in that time?

I thought he was doing a supertask with his sweaters…

Man I sure do love knowledge

He actually just asked us the same question about 5 or 6 times while retaining viewer attention. Fascinating

Michael : Cuts the cake in 2 pieces

After 2 days:

Michael: Cuts the cake in 2 pieces

After 2000 years:

Michael : searched on google how to cut an atom in half

Me: '_' bruh that's ilegal :/

Fake

9:37 Reminds me if that GE lightbulb reset add XD

The solution; the lamp breaks.