Shadowing Practice: The Banach–Tarski Paradox - Learn English Speaking with Video
Creating lesson...
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Hey Vsauce, Michael here.
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There's a famous way to seemingly create chocolate out of nothing.
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Maybe you've seen it before.
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This chocolate bar is four squares by eight squares.
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But if you cut it like this, and then like this, and finally like this,
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you can rearrange the pieces like so and wind up with the same four by eight bar,
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but with a leftover piece apparently created out of thin air.
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There's a popular animation of this illusion as well.
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I call it an illusion because it's just that, fake.
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In reality, the final bar is a bit smaller.
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It contains this much less chocolate.
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Each square along the cut is shorter than it was in the original, but the cut makes it difficult to notice right away.
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The animation is extra misleading because it tries to cover up its deception.
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The lost height of each Each square is surreptitiously added in while the piece moves to make it hard to notice.
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I mean, come on.
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Obviously, you cannot cut up a chocolate bar and rearrange the pieces into more than you started with.
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- Okay.
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Or can you?
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One of the strangest theorems in modern mathematics is the Banach -Tarski paradox.
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It proves that there is in fact a way to take an object and separate it into five different pieces. of the original item.
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Same density, same size, same everything.
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Seriously, to dive into the mind blow that it is and the way it fundamentally questions math and ourselves,
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we have to start by asking a few questions.
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First, what is infinity?
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A number?
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I mean, it's nowhere on the number line, but we often say things like, "There's an infinite number of blah blah blah.
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And as far as we know, infinity could be real.
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The universe may be infinite in size and flat, extending out forever and ever without end,
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beyond even the part we can't observe or ever hope to observe.
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That's exactly what infinity is.
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Not a number per se, but rather a size.
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the size of something that doesn't end.
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Infinity is not the biggest number.
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Instead, it is how many numbers there are.
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But there are different sizes of infinity.
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The smallest type of infinity is countable infinity, the number of hours in forever.
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It's also the number of whole numbers that there are, natural numbers, the numbers we use when counting things like 1,
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2, 3, 4, 5, 6, and so on.
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Sets like these are unending, but they are countable.
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Countable means that you can count them from one element to any other in a finite amount of time.
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Even if that finite amount of time is longer than you will live, or the universe will exist for, it's still finite.
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Uncountable infinity, on the other hand, is literally bigger.
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Too big to even count.
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The number of real numbers that there are, not just whole numbers, but all numbers, is uncountably infinite.
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You literally cannot count even from zero to one in a finite amount of time by naming every real number in between.
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I mean, where do you even start?
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Zero?
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Okay, but What comes next?
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0 .000000.
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Eventually, we would imagine a 1 going somewhere at the end, but there is no end.
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We could always add another 0.
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Uncountability makes this set so much larger than the set of all whole numbers
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that even between 0 and 1 there are more numbers than there are whole numbers on the entire endless number line.
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Derokantor's famous diagonal argument helps illustrate this.
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Imagine listing every number between 0 and 1.
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Since they are uncountable and can't be listed in order, let's imagine randomly generating them forever with no repeats.
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Bye.
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Each number we generate can be paired with a whole number.
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If there's a one -to -one correspondence between the two, that is if we can match one whole number to each real number on our list,
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that would mean that countable and uncountable sets are the same size.
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But we can't do that.
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Even though this list goes on forever, Forever isn't enough.
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Watch this.
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If we go diagonally down our endless list of real numbers and take the first decimal of the first number, and the second of the second number,
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the third of the third, and so on, and add one to each, subtracting one if it happens to be a nine,
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we can generate a new real number that is obviously between 0 and 1,
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but since we've defined it to be different from every number on our endless list in at least one place,
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It's clearly not contained in the list.
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In other words, we've used up every single whole number, the entire infinity of them,
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and yet we can still come up with more real numbers.
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Here's something else that is true but counterintuitive.
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There are the same number of even numbers as there are even and odd numbers.
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At first, that sounds ridiculous.
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Clearly there are only half as many even numbers as all whole numbers.
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But that intuition is wrong.
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The set of all whole numbers is denser, but every even number can be matched with a whole number.
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You will never run out of members of either set, so this one -to -one correspondence shows that both sets are the same size.
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In other words, infinity divided by two is still infinity.
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Infinity plus one is also infinity. of the Grand Hotel.
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Imagine a hotel with a countably infinite number of rooms.
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But now, imagine that there is a person booked into every single room.
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Seemingly, it's fully booked.
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Right?
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No. Infinite sets go against common sense.
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You see, if a new guest shows up and wants a room, all the hotel has to do is move the guest in room number one to room number two.
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And the guest in room two to room three and three to four and four to five and so on.
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Because the number of rooms is never ending, we cannot run out of rooms.
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Infinity minus one is also infinity again.
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If one guest leaves the hotel, we can shift every guest the other way.
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Guest two goes to room one, three to two, four to three, and so on.
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Because we have an infinite amount of guests, that is a never -ending supply of them, no room will be left empty As it turns out,
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you can subtract any finite number from infinity and still be left with infinity.
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It doesn't care.
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It's unending.
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Banoktarski hasn't left our sights yet.
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All of this is related.
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We are now ready to move on to Shapes.
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Hilbert's Hotel can be applied to a circle.
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Points around the circumference can be thought of as guests.
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If we remove one point from the circle, That point is gone.
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Right?
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Infinity tells us it doesn't matter.
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The circumference of a circle is irrational.
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It's the radius times 2 pi.
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So if we mark off points beginning from the hole, every radius length along the circumference going clockwise,
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we will never land on the same point twice.
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Ever.
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has checked out, we can just shift the rest, move them counterclockwise, and every room will be filled.
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Point 1 moves to fill in the hole.
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Point 2 fills in the place where point 1 used to be.
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3 fills in 2 and so on.
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Since we have an unending supply of numbered points, No hole will be left unfilled.
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The missing point is forgotten.
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we apparently never needed it to be complete.
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There's one last neato consequence of infinity we should discuss before tackling Bonnachtarski.
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Ian Stewart famously proposed a brilliant dictionary, one that he called the Hyper Webster.
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The Hyper Webster lists every single possible word of any length formed from the 26 letters in the English alphabet.
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It begins with A followed by AA, then AAA, then AAAA, and after an infinite number of those,
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AB then ABA, then ABAA, ABAAA, and so on until Z.
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ZA, ZAA, etc, etc, until the final entry, an infinite sequence of Zs.
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Such a dictionary would contain every single word.
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Nay, every single thought, definition, description, truth, lie, name, story, what happened to Amelia Earhart would be in that dictionary.
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as well as every single thing that didn't happened to Amelia Earhart.
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everything that could be said using our alphabet.
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Obviously, it would be huge.
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But the company publishing it might realize that they could take a shortcut.
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If they put all the words that begin with "A" in a volume titled "A", they wouldn't have to print the initial "A".
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Readers would know to just add the A because it's the A volume.
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By removing the initial A, the publisher is left with every A word Sans the first A.
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which has surprisingly become every possible word.
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Just one of the 26 volumes has been decomposed into the entire thing.
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It is now.
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that we're ready to investigate this video's titular paradox.
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What if we turned an object, a 3D thing, into a Hyper Webster?
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Could we decompose pieces of it into the whole thing?
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Yes.
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The first thing we need to do is give every single point on the surface of the sphere, one name and one name only.
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A good way to do this is to name them after how they can be reached by a given starting point.
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If we move this starting point across the surface of the sphere in steps that are just the right length,
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no matter how many times or in what direction we rotate, So long as we never backtrack, it will never wind up in the same place twice.
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We only need to rotate in four directions to achieve this paradox: up, down, left and right, around two perpendicular axes.
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We are going to need every single possible sequence
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that can be made of any finite length out of just these four rotations.
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That means we will need left, right, up and down, as well as left left,
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left up, left down, but of course not left right because, well, that's backtracking.
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Going left and then right means you're the same as you were before you did anything.
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So, no left rights, no right lefts, and no up downs, and no down ups.
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Also, notice that I'm writing the rotations in order right to left.
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So the final rotation is the leftmost letter.
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That will be important later on.
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Anyway, a list of all possible sequences of allowed rotations that are finite in length, is Well, huge.
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countably infinite, in fact.
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But if we apply each one of them to a starting point, in green here, and then name the point we land on after the sequence that brought us there,
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we can name a countably infinite set of points on the surface.
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Let's look at how, say, these four strings on our list would work.
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right up left.
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Okay, rotating the starting point this way takes us here.
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Let's color code the point based on the final rotation in its string.
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In this case, it's left and and for that we will use purple.
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Next up, "Down, down." that sequence takes us here.
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We name the point DD and color it blue since we ended with a down rotation.
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RDR, that will be this point's name, takes us here, and for a final right rotation, let's use red.
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Finally, for a sequence that ends with "up", let's color code the point.
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Orange.
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Now, if we imagine completing this process for every single sequence, we will have a countably infinite number of points named and color -coded.
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That's great!
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But not enough there are an uncountably infinite number of points on a spheres.
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surface.
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But no worries, we can just pick a point we missed.
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any point and color it green, making it a new starting point, and then run every sequence from here.
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After doing this, to an uncountably infinite number of starting points, we will have indeed named and colored every single point on the surface.
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just once.
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with the exception of Poles.
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Every sequence has two poles of rotation, locations on the sphere that come back to exactly where they started.
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for any sequence of right, or left rotations, the poles are the north and south poles.
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The problem with poles like these is
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that more than one sequence can lead us to them they can be named more than once
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and be colored in more than one color.
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For example, if you follow some other sequence to the North or South Pole, any subsequent rights or lefts will be equally valid names.
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In order to deal with this, we're going to just count them out of the normal scheme and color them all yellow.
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Every sequence has two, so there are a countably infinite amount of them.
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Now, with every point on the sphere given just one name, in just one of six colors we are ready to take the entire sphere apart.
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Every point on the surface corresponds to a unique line of points below it,
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all the way to the center point and we will be dragging every points line along with it.
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The lone center point we will set aside.
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Okay, first we cut out and extract all the yellow poles.
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the green starting points, the orange up points the blue down points, and the red and purple left and right points.
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That's the entire sphere.
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With just these pieces, you could build the whole thing.
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But take a look at the left piece.
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It is defined by being a piece composed of every point accessed via a sequence ending with a left rotation.
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If we rotate this piece right, that's the same as adding an R to every point's name.
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But left and then right is a backtrack.
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They cancel each other out.
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And look what happens when we reduce them away.
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The set becomes the same as a set of all points with names that end with L,
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but also U, D, and every point reached with no rotation.
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That's the full set of starting points.
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We have turned less than a quarter of the sphere into nearly three quarters.
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Just by rotating it, we added nothing.
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It's like the Hyper Webster.
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If we add the right piece and the poles of rotation in the center point, well, we've got the entire sphere again, but with stuff left over.
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To make a second copy, let's rotate the up piece down.
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The down -ups cancel because, well, it's the same as going nowhere, and we're left with a set of all starting points,
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the entire up piece, the right piece and the left piece But there's a problem here.
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We don't need this extra set of starting points.
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We still haven't used the original ones.
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No worries.
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Let's just start over we can just move everything from the up piece that turns into a starting point when rotated down.
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That means every point whose final rotation is up.
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Let's put them in the down piece.
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Of course, after rotating, points named UU will just turn into points named U.
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And that would give us a copy here and here.
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So, as it turns out, we need to move all points with any name that is just a string of U's.
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We will put them in the down piece and rotate the up piece down, which makes it congruent to the up, right, and left pieces.
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Add in the down piece along with some up and the starting point piece and, well, we're almost done.
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Rotation and center are missing from this copy, but no worries.
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There's a countably infinite number of holes where the poles of rotation used to be.
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Which means there is some pole around which we can rotate this sphere such that every pole hole orbits around
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without hitting another.
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Well, this is just a bunch of circles with one point missing.
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We fill them each like we did earlier.
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And we do the same for the center point.
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Imagine a circle that contains it inside the sphere and just fill in from infinity.
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And look what we've done.
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We have taken one sphere, and turned it into two identical spheres without adding Anything.
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1 plus 1 equals 1.
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Thank you.
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That took a while to go through, but the implications are huge and mathematicians, scientists, and philosophers are still debating them.
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Could such a process happen in the real world I mean, it can happen mathematically, and math allows us to abstractly predict
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and describe a lot of things in the real world with amazing accuracy.
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But does the Banach -Tarski paradox take it too far?
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Is it a place where math and physics Separate.
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We still don't know.
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History is full of examples of mathematical concepts developed in the abstract that we
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did not think would ever apply to the real world for years,
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decades, centuries, until Eventually, science caught up and realized they were totally applicable and useful,
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The Bonnachtarski paradox could actually happen in our real world.
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The only catch, of course, is that the five pieces you cut your object into aren't simple shapes,
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they must be infinitely complex and detailed That's not possible to do in the real world, where measurements can only get so small,
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and there's only a finite amount of time to do anything.
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but math says it's theoretically valid and some scientists think it may be physically valid 2.
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There have been a number of papers published suggesting a link between Banach Tarski,
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and the way tiny, tiny subatomic particles can collide.
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high energies and turn into more particles than we began with.
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We are finite creatures.
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Our lives are small, and can only scientifically consider a small part of reality,
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What's common for us is just a sliver of what's available.
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We can only see so much of the electromagnetic spectrum.
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We can only delve so deep into extensions of space.
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Common sense applies to that which we can access.
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But common sense is just that.
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"common." If total sense is what we want, we should be prepared to accept that we shouldn't call infinity weird or strange.
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The results we've arrived at by accepting it are valid, true within the system we use to understand, measure, predict, and order the universe.
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Perhaps the system still needs perfecting.
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But at the end of the day, history continues to show us that the universe isn't strange.
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We are.
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And as always, thanks for watching there that really helped me wrap my mind kind of, around Banat Tarski.
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First of all, Leonard Wapner's The Pea and The Sun.
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This book is fantastic and it's full of a lot of the preliminaries needed to understand the proof that comes later.
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He also talks a lot about the ramifications of what Bonnachtarski and their theorem might mean for mathematics.
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Also, if you want to talk about math and whether it's discovered or invented, whether it really truly will map onto the universe, .
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Yanofsky's...
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The Outer Limits of Reason is great.
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This is the favorite book of mine that I've read this entire year.
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Another good one is E.
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Brian Davies.
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Why Beliefs Matter.
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This is actually Korn's favorite book as you might be able to see there.
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It's delicious and full of lots of great information about the limits of what we can know, and what science is, and what mathematics is.
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If you love infinity and math, I cannot more highly recommend Matt Parker's Things to Make and Do in the Fourth Dimension.
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He is hilarious and this book is very very great at explaining some pretty awesome things.
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So, Keep reading and if you're looking for something to watch, I hope you've already watched Kevin Lieber's film on Field Day.
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I already did a documentary about Whittier, Alaska over there.
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Kevin's got a great short film about putting things out on the internet and having people react to them.
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There's a rumor that Jake Roper might be doing something on Field Day soon, so check out mine, check out Kevin's,
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and subscribe to Field Day for upcoming Jake Roper action, yeah?
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He's actually in this room right now.
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Say hi, Jake.
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Hi.
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Thanks for filming this by the way.
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Guys, I really appreciate
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who you all are, and as always, thanks for watching.
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Why practice speaking with this video?
Practicing speaking with this video on the Banach-Tarski Paradox can significantly enhance your English language skills. The complex language and mathematical concepts presented provide a unique opportunity to engage with advanced vocabulary, making it an excellent resource for learners aiming to improve their English pronunciation. By using a technique called shadow speaking, you can mimic the speaker's intonation, pace, and pronunciation, which will help you internalize the sounds and rhythms of English. Additionally, the engaging topic can boost your confidence and encourage you to discuss abstract ideas in English, which is crucial for fluency. As you repeat phrases and sentences, your listening skills will also improve, allowing for better comprehension in future conversations.
Grammar & Expressions in Context
The video features several advanced structures that can be beneficial for learners looking to enhance their expressive capabilities in English:
- Conditional Clauses: Phrases such as “If we remove one point from the circle...” showcase how to form conditional sentences, which are essential for discussing hypothetical situations.
- Passive Voice: The use of passive constructions like “is defined” demonstrates a technique for emphasizing actions or states over the subjects performing them, helping learners convey information succinctly.
- Comparative Structures: Expressions such as “even bigger” and comparisons regarding infinity help articulate contrasts, which can clarify complex ideas in discussions.
- Modals of Speculation: Phrases like “we could always add another 0” provide examples of how to express possibility, which is useful for conjecturing in conversation.
Common Pronunciation Traps
As you engage with the video, be mindful of certain pronunciation challenges that may arise:
- “Paradox”: This word can be tricky; make sure to emphasize the second syllable, pronouncing it as “pair-uh-dox.” Practicing with shadow speech can help you get it right.
- “Uncountable”: A common mistake is misplacing stress. It should be pronounced “un-COUNT-able,” with stronger emphasis on the second syllable.
- “Infinity”: This term should be articulated clearly, breaking it down to “in-FIN-i-tee,” to avoid slurring sounds together.
- “Georg Cantor”: Ensure you pronounce the name correctly, focusing on the soft “g” sound as in “GE-org.” This name is particularly valuable as referencing figures in discussions about mathematics can enhance your vocabulary.
By paying attention to these nuances and honing your pronunciation skills, you’ll be on your way to clearer and more confident English communication through effective shadowing techniques.
Grammar in this video
The structures the speaker uses most, with the exact words from the video:
| Structure | In the video |
|---|---|
| “Used to” used to + verb — a past habit or state that is no longer true | used to be |
| Present perfect have/has + past participle — a past action that still matters now | you've seen · we've defined · hasn't left |
| Passive voice be + past participle — the focus is on what happens, not who does it | can't be listed · can be paired · can be matched |
What is the Shadowing Technique?
Shadowing is a science-backed language learning technique originally developed for professional interpreter training and popularized by polyglot Dr. Alexander Arguelles. The method is simple but powerful: you listen to native English audio and immediately repeat it out loud — like a shadow following the speaker with just a 1–2 second delay. Unlike passive listening or grammar drills, shadowing forces your brain and mouth muscles to simultaneously process and reproduce real speech patterns. Research shows it significantly improves pronunciation accuracy, intonation, rhythm, connected speech, listening comprehension, and speaking fluency — making it one of the most effective methods for IELTS Speaking preparation and real-world English communication.