シャドーイング練習: The Banach–Tarski Paradox - 動画で英語スピーキングを学ぶ

Hey Vsauce, Michael here.

There's a famous way to seemingly create chocolate out of nothing.

Maybe you've seen it before.

This chocolate bar is four squares by eight squares.

But if you cut it like this, and then like this, and finally like this,

you can rearrange the pieces like so and wind up with the same four by eight bar,

but with a leftover piece apparently created out of thin air.

There's a popular animation of this illusion as well.

I call it an illusion because it's just that, fake.

In reality, the final bar is a bit smaller.

It contains this much less chocolate.

Each square along the cut is shorter than it was in the original, but the cut makes it difficult to notice right away.

The animation is extra misleading because it tries to cover up its deception.

The lost height of each Each square is surreptitiously added in while the piece moves to make it hard to notice.

I mean, come on.

Obviously, you cannot cut up a chocolate bar and rearrange the pieces into more than you started with.

- Okay.

Or can you?

One of the strangest theorems in modern mathematics is the Banach -Tarski paradox.

It proves that there is in fact a way to take an object and separate it into five different pieces. of the original item.

Same density, same size, same everything.

Seriously, to dive into the mind blow that it is and the way it fundamentally questions math and ourselves,

we have to start by asking a few questions.

First, what is infinity?

A number?

I mean, it's nowhere on the number line, but we often say things like, "There's an infinite number of blah blah blah.

And as far as we know, infinity could be real.

The universe may be infinite in size and flat, extending out forever and ever without end,

beyond even the part we can't observe or ever hope to observe.

That's exactly what infinity is.

Not a number per se, but rather a size.

the size of something that doesn't end.

Infinity is not the biggest number.

Instead, it is how many numbers there are.

But there are different sizes of infinity.

The smallest type of infinity is countable infinity, the number of hours in forever.

It's also the number of whole numbers that there are, natural numbers, the numbers we use when counting things like 1,

2, 3, 4, 5, 6, and so on.

Sets like these are unending, but they are countable.

Countable means that you can count them from one element to any other in a finite amount of time.

Even if that finite amount of time is longer than you will live, or the universe will exist for, it's still finite.

Uncountable infinity, on the other hand, is literally bigger.

Too big to even count.

The number of real numbers that there are, not just whole numbers, but all numbers, is uncountably infinite.

You literally cannot count even from zero to one in a finite amount of time by naming every real number in between.

I mean, where do you even start?

Zero?

Okay, but What comes next?

0 .000000.

Eventually, we would imagine a 1 going somewhere at the end, but there is no end.

We could always add another 0.

Uncountability makes this set so much larger than the set of all whole numbers

that even between 0 and 1 there are more numbers than there are whole numbers on the entire endless number line.

Derokantor's famous diagonal argument helps illustrate this.

Imagine listing every number between 0 and 1.

Since they are uncountable and can't be listed in order, let's imagine randomly generating them forever with no repeats.

Bye.

Each number we generate can be paired with a whole number.

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,

that would mean that countable and uncountable sets are the same size.

But we can't do that.

Even though this list goes on forever, Forever isn't enough.

Watch this.

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,

the third of the third, and so on, and add one to each, subtracting one if it happens to be a nine,

we can generate a new real number that is obviously between 0 and 1,

but since we've defined it to be different from every number on our endless list in at least one place,

It's clearly not contained in the list.

In other words, we've used up every single whole number, the entire infinity of them,

and yet we can still come up with more real numbers.

Here's something else that is true but counterintuitive.

There are the same number of even numbers as there are even and odd numbers.

At first, that sounds ridiculous.

Clearly there are only half as many even numbers as all whole numbers.

But that intuition is wrong.

The set of all whole numbers is denser, but every even number can be matched with a whole number.

You will never run out of members of either set, so this one -to -one correspondence shows that both sets are the same size.

In other words, infinity divided by two is still infinity.

Infinity plus one is also infinity. of the Grand Hotel.

Imagine a hotel with a countably infinite number of rooms.

But now, imagine that there is a person booked into every single room.

Seemingly, it's fully booked.

Right?

No. Infinite sets go against common sense.

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.

And the guest in room two to room three and three to four and four to five and so on.

Because the number of rooms is never ending, we cannot run out of rooms.

Infinity minus one is also infinity again.

If one guest leaves the hotel, we can shift every guest the other way.

Guest two goes to room one, three to two, four to three, and so on.

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,

you can subtract any finite number from infinity and still be left with infinity.

It doesn't care.

It's unending.

Banoktarski hasn't left our sights yet.

All of this is related.

We are now ready to move on to Shapes.

Hilbert's Hotel can be applied to a circle.

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

220

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.

228

Thank you.

229

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

231

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,

237

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,

240

they must be infinitely complex and detailed That's not possible to do in the real world, where measurements can only get so small,

241

and there's only a finite amount of time to do anything.

242

but math says it's theoretically valid and some scientists think it may be physically valid 2.

243

There have been a number of papers published suggesting a link between Banach Tarski,

244

and the way tiny, tiny subatomic particles can collide.

245

high energies and turn into more particles than we began with.

246

We are finite creatures.

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Our lives are small, and can only scientifically consider a small part of reality,

248

What's common for us is just a sliver of what's available.

249

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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このレッスンについて

このレッスンでは、バナッハ–タルスキーの逆説について深く探求します。この逆説は、無限の概念と、いかにして数学が直感に反することがあるかを理解する助けになります。学習者は、この内容を通じて、数学的な思考の重要性や、無限のサイズ、そしてそれが日常生活にどのように関連しているかについて考えることができるでしょう。

重要な語彙とフレーズ

逆説 (ぎゃくせつ) - 一見矛盾するように思えるが、実際には真実である状況。
無限 (むげん) - 終わりがないこと。無限の概念を理解することが重要です。
幾何学 (きかがく) - 形や空間を研究する数学の一分野。
範囲 (はんい) - 特定の数の範囲、無限の考えを考察する際に使われます。
対称性 (たいしょうせい) - 形や構造の均等さや調和。
数えられる (かぞえられる) - 数を数えることができる無限のセット。
非数えられる (ひかぞえられる) - 数えきれない無限のセット。
幾何学的な図形 (きかがくてきなずけい) - 図形や形状を表す場合に使用される。

練習のコツ

この動画のスピードとトーンは、英語の発音を良くするためのシャドーイングに特に効果的です。特に、最初はゆっくりとしたペースで音声を聴き、その後、少しずつスピードを上げていくことが大切です。自分が聴いた内容を繰り返す際には、以下のポイントに注意しましょう：

まず、動画の内容を少しずつ聴き、意図的に発音とリズムを模倣します。
重要なフレーズや言葉を何度も繰り返し、自分の口を動かすことで身体に馴染ませます。
絶え間なく続けていけるように、自分のペースで無理なく続けることが大切です。
特に、バナッハ–タルスキーの逆説に関連する部分で、どのように思考が進むかを意識しながらシャドーイングを行います。
英語のシャドーイングを実践する際には、リスニングスキルも同時に向上することを意識しましょう。

このようにして、IELTS スピーキング対策をしながら、英語の発音を良くし、さらに自信を持って会話ができるようになるでしょう。shadowing siteやshadow speakを利用して、さらなる練習を進めてみてください。

この動画の文法

話し手がよく使っている文型を、動画の実際の表現とともに紹介します。

文型	動画での表現
「used to」 used to + 動詞 — 今はもう違う過去の習慣や状態	used to be
現在完了形 have/has + 過去分詞 — 過去の出来事が今も関係している	you've seen · we've defined · hasn't left
受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる	can't be listed · can be paired · can be matched

シャドーイングとは？英語上達に効果的な理由

シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。

シャドーイングのやり方: ステップ別の完全ガイドを読む →

シャドーイング練習: The Banach–Tarski Paradox - 動画で英語スピーキングを学ぶ

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このレッスンについて

重要な語彙とフレーズ

練習のコツ

この動画の文法

シャドーイングとは？英語上達に効果的な理由