The Banach–Tarski Paradox

Hey, vsauce. Michael here. There is a famous way of creating chocolate out of nowhere. You may have seen it before. This chocolate bar is 4 squares by 8 squares, but if you cut it like this and then so and finally, you can reorganize the pieces as and end the same bar of 4 by 8 but with a leftover piece, apparently created from nothing. There is also a popular animation of this illusion. I call it an illusion because it's just that. Fake. Actually, the final bar is a bit smaller. It contains much less chocolate. Each square along the cut is shorter than it was in the original, but the cut hinders the notification immediately.

The animation is more misleading, because he tries to cover up his deception. The lost height of each square is added surreptitiously while the piece moves to be difficult to notice. I mean, come on, you can't obviously cut a chocolate bar and reorganize the pieces in more than you started. Or can you? One of the strangest theorems in modern mathematics is Banach-Tarski's

paradox

. It shows that, in fact, there is a way of taking an object and separating it into 5 different pieces. And then, with those five pieces, simply reorganize them. Stretch is not required in two exact copies of the original article.

The same density, same size, everything. Oh really. To immerse ourselves in the mind that is and the way in which mathematics questions fundamentally and ourselves, we have to start asking some questions. First, what is infinity? A number? I mean, it is nowhere in the numerical line, but we often say things as if there were an infinite "Blah-Bla" number ". And as far as we know, infinity could be real. The universe can be infinite in size and plane, which extends forever without end, beyond the part that we can observe or hopes to observe. That is exactly what Infinity is.

It is not a number per se, but rather a size. The size of something that does not end. Infinity is not the largest number, however, it is how many numbers there are. But there are different infinity sizes. The smallest type of infinity is the infinity accounting. The number of hours in eternity. It is also the number of whole numbers that are, natural number, the numbers we use when counting things, such as 1, 2, 3, 4, 5, 6, etc. Sets like these are endless, but they are accounting. Accounting means that you can count them from one element to any other in a finite time, even if that finite amount of time is longer than it will live or the universe will exist, it is still finite.

Incadential Infinity, on the other hand, is literally larger. Too big to count. The number of real numbers that exist, not only whole numbers, but all the numbers is countless infinity. Literally, you cannot count even from 0 to 1 in a finite time by naming each real number in the middle. I mean, where do you start? Zero, it's fine. But what comes next? 0.000000 ... Eventually, we imagine that a 1 is going somewhere at the end, but there is no end. We could always add another 0. Incontability makes this set much larger than the whole of all whole numbers than even between 0 and 1, there are more numbers than the whole numbers throughout the endless numerical line.

The famous diagonal argument of Georg Cantor helps to illustrate this. Imagine listing each number between zero and one. Since they are countless and cannot be listed in order, let's imagine randomly generating them without repetitions. Each regeneration number can be paired with a complete number. If there is a correspondence of one between the two, that is, if we can match a complete number with each real number on our list, that would mean that the accounting and countless sets are of the same size. But we cannot do that, although this list continues forever. Forever is not enough. Look at this.

If we lower our endless list of real numbers diagonally and take the first decimal of the first issue and the second of the second issue, the third of the third and so on and add one to each one, subtracting one if it turns out to be a nine, we can generate a new real number that is obviously between 0 and 1, but since we define that it is different from each number in our endless list and at least a real number, clearly, it is clearly not between the list. In other words, we have used each whole number, all the infinity of them and, nevertheless, we can still find more real numbers.

There are more than is true but contra-intuitive. There are the same number of even numbers that there are even and odd numbers. At first, that sounds ridiculous. Clearly, there are only half of the even numbers that all whole numbers, but that intuition is incorrect. The set of all complete numbers is denser, but each uniform number can coincide with an integer. It will never remain without members, so this correspondence of one shows that both sets are the same size. In other words, the infinite divided by two remains infinite. Infinity plus one is also infinite. A good illustration of this is Hilbert's

paradox

at the Grand Hotel.

Imagine a hotel with a contaminally infinite number of rooms. But now, imagine that there is a reserved person in each room. Apparently, he is completely reserved, right? No. The infinite sets go against common sense. You will see, if a new guest appears and want a room, all that the hotel has to do is move the guest in room number 1 to number 2. And a guest in room 2 to room 3 and 3 to 4 and 4 to 5, and so on. Because the number of rooms never ends, we cannot run out of rooms. Infinity -1 is also infinite again. If a guest leaves the hotel, we can change each guest for the other side.

Guest 2 goes to room 1, 3 to 2, 4 to 3 and so on, because we have an infinite amount of guests. That is an endless supply of them. There will be no empty space. It turns out that any finite number of Infinity can subtract and still keep Infinity. He doesn't care. It is endless. Banach-Tarski has not yet left our view. All this is related. Now we are ready to move on to shapes. The Hilbert hotel can be applied to a circle. Points around the circumference can be considered as guests. If we eliminate a point from the circle, that point has disappeared, right?

Infinity tells us that it doesn't matter. The circumference of a circle is irrational. It is the Radio Times 2pi. So, if we mark the points that start from the set, each radio length along the circumference that goes in the direction of the clock needles, we will never land at the same point twice, never. We can count every point that we mark with a complete number. So this set is endless, but accounting, like guests and rooms at the Hilbert hotel. And like those guests, although one has been reviewed, we can change the rest. Move them in an anti -Horary sense and each room will fill 1 moves to fill the hole, point 2 fills the place where point 1, 3 are filled in 2 and so on.

Since we have an endless supply of numbered points, no hole will be left. The missing point is forgotten. Apparently we never need to be complete. There is a last consequence of the infinite that we must discuss before addressing Banach-Tarski. Ian Stewart proposed a brilliant dictionary. One that called the hyperwebster. The hyperwebster lists each possible word of any length formed from the 26 letters in the English alphabet. Start with "A", followed by "AA", then "AAA", then "AAAA". And after an infinite number of those, "ab", then "aba", then "abaa", "abaaa", and so on to "z", za "," zaa ", etc., etc., until the final entry, the story, the name, the name, the story of" z ". which has become all possible words. departure.

The center is missing in this copy, but do not worry. It contains within the sphere and simply complete from infinity and look at what we have done. Mathematics allow us to predict in an abstract way and describe many things in the real world with surprising precision, but does the Banach-Tarski paradox take it too far? Science was updated and realized that they were fully applicable and useful. They say that it is theoretically valid and some scientists think that it can also be physically valid. of what is available. predict and order the universe. Wapner My favorite book that I have read this year.

This is actually Corn's favorite book, as you could see there. It is delicious and is full of much information about the limits of what we can know and what science is and what mathematics is. If you love infinity and mathematics, I can not recommend the "things to do and do in the fourth dimension of Matt Parker. It's funny and this book is very good to explain some incredible things. So keep reading, and if you're looking for something to see, I hope you have already seen Kevin Lieber's movie on the field day. I already made a documentary about Whittier, Alaska there.

Short film about putting things on the Internet and making people react to them.This moment, greets Jake. Hello. Thanks for filming this, by the way. Boys, I really appreciate who all are. And as always, thanks for looking.