Mathematics: Sizes of infinity part 1: Hilbert's Hotel

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Mathematics: Sizes of infinity part 1: Hilbert's Hotel발음듣기

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(Intro music) Hi, I am Agustin Rayo.발음듣기

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I'm an associate professor of philosophy at MIT, and today I want to talk about how there are some infinities that are bigger than others.발음듣기

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So, a first exercise to see why some infinities are bigger than others is to think about Hilbert's Hotel.발음듣기

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So Hilbert's Hotel is like an ordinary hotel, except that instead of having finitely many rooms like most hotels, it has infinitely many rooms.발음듣기

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So we can draw this. We're just having a very long rectangle with lots of rooms, and we're going to number each room with a natural number.발음듣기

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So, the first natural number is zero, after that comes one and two and three and four and five and so forth.발음듣기

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And I'm going to assume that this hotel is completely full.발음듣기

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So in other words, every room has a person inside it.발음듣기

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So we're going to draw a person inside each room.발음듣기

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And I'm going to assume that these various persons are numbered, too.발음듣기

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So maybe each of them is holding a piece of paper with a number.발음듣기

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The person in room zero has a piece of paper with zero on it, the person in room one has a piece of paper with the number one in it, and so forth.발음듣기

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So now ask yourself this question: what would happen if an extra person were to come to the hotel?발음듣기

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Could they be accommodated? Now, in a finite hotel, the answer is "No," because all the rooms are already occupied, and we can assume that these are prickly guests and don't want to share rooms.발음듣기

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But in an infinite hotel, it can be done.발음듣기

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So think of it this way.발음듣기

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We can just ask each person to move one room to the right.발음듣기

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Mr. Zero, who's now in room zero, moves to room one, and Miss One, who's now in room one, moves to room two, and two moves to three, and three moves to four, and four moves to five, and so forth.발음듣기

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So then the result is that all of our original guests are in rooms, but our first room is vacant, because now there's nobody in room zero, so we can welcome our additional guest.발음듣기

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One thing that's amazing and worth emphasizing is that the hotel was full at the beginning, and we could still fit in more people.발음듣기

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That's amazing, but true. Part of the reason that's possible is that there are two things that come together in the finite case, but need to be kept apart in the infinite case.발음듣기

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So in the finite case, if you have a set and you add some extra things to the set, the set you get is bigger than the original one, but in the infinite case, that's not necessarily true, because have your original set just consist of the original hotel guests, and then add one more.발음듣기

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The new set isn't bigger, in the sense that you can still put each element of the new set into a different room in your hotel.발음듣기

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So if you measure size in terms of how big a hotel you would need to accommodate the members of the set, then we need to conclude that the set of the original guests, and the set of the original guests plus one, are of the same size.발음듣기

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Crazy! It's this amazing thing. And if you think of it, you can see that we could have accommodated any finite number of new guests, too, because suppose we get seven hundred new guests.발음듣기

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Then we can just ask each of our original guests to move seven hundred rooms to the right.발음듣기

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But wait! What if we got infinitely many new guests?발음듣기

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OK, so suppose that we have infinitely many new guests, and they too are numbered with the natural numbers, so the first one is holding zero, and the second one is holding a one, and the next one is holding the two, and so forth.발음듣기

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How could we accommodate them? Well, we can do it like this.발음듣기

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We just ask each of our original guests to move to the result of multiplying their current room by two.발음듣기

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So the guy in zero goes to two times zero equals zero, so he stays.발음듣기

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The person in one moves to one times two, that's two, so she moves one to the right.발음듣기

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The person in two moves to two times two, that's four.발음듣기

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She moves two to the right, and so forth.발음듣기

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So the result of this is that all the old guests are occupying even-numbered rooms, and all of the odd-numbered rooms are free, so we can ask each of the new guests to go to an odd-numbered room.발음듣기

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Which one? They can just look at their number, multiply it by two, and add one, and they can go to that room.발음듣기

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So here's the lesson of this.발음듣기

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The lesson of this is that if we have an infinite hotel, we can accommodate one copy of the natural numbers, but we can also accommodate two copies of the natural numbers, and in fact, we can accommodate as many copies of the natural numbers as there are natural numbers.발음듣기

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But before tackling that question, it's useful to consider a different one, and this is the first of two wonderful theorems that were proved by Georg Cantor in the nineteenth century.발음듣기

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The first theorem is that you can put the rational numbers in one-one correspondence with the natural numbers.발음듣기

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Now, what is a rational number?발음듣기

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A rational number is a number of the form "a/b," where "a" and "b" are both natural numbers, and we assume that "b" is not zero.발음듣기

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So what Cantor did is show that we can pair up the natural numbers with the rational numbers with no remainder.발음듣기

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Every natural number gets a unique rational number, and every rational number gets a unique natural number and nobody's left out.발음듣기

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So here's how to do it.발음듣기

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We simply draw a matrix. Each column is going to correspond to a numerator, and each row is going to correspond to a denominator.발음듣기

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So this is how you fill out the matrix.발음듣기

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And so forth. So you can see that every rational number is on this matrix, because remember, you get a rational number by taking a natural number and dividing it by a natural number different from zero.발음듣기

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Suppose your rational number is seventeen divided by ninety-four.발음듣기

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In order to find the cell that corresponds to that number, just go to the column that's labeled "seventeen," and to the row that's labeled "ninety-four," and that's where your number is going to be.발음듣기

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So every rational number is on the matrix.발음듣기

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So what Cantor discovered is that you can assign a natural number to each cell, and here's how to do it.발음듣기

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Just diagonalize. So have this one be zero, and this one, one.발음듣기

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And two, and three, and four, and five, and six, and seven, and eight, and so forth.발음듣기

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Using this system, every rational number got assigned a natural number.발음듣기

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So now we can come back to our hotel.발음듣기

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So remember, our question was "How could you fit infinitely many infinities into your hotel?"발음듣기

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Well, now we have a way of doing it.발음듣기

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So, think of each column on our matrix as corresponding to one infinity of new guests.발음듣기

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In order to decide which room to send a person to, all you need to do is use Cantor's trick.발음듣기

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So, assign a natural number to each cell in the matrix, and ask the person to go to the room corresponding to that number.발음듣기

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So there, we fit infinitely new infinities of guests into our hotel. Amazing!발음듣기

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