To Infinity and Beyond To Infinity and Beyond 1/43
Infinity The concept of infinity has both fascinated and frustrated people for millennia. We will discuss some historical problems about infinity, some modern (around 100 years old) ideas about infinity, and some interesting puzzles about the concept. To Infinity and Beyond 2/43
Zeno s Paradoxes There are a series of paradoxes possibly coming from the Greek philosopher Zeno (490-430 BC). One we will discuss is the paradox of Achilles and the Tortoise. Aristotle is quoted referring to this paradox: In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. To Infinity and Beyond 3/43
Here is the statement of the paradox (borrowed in large part from Wikipedia): In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Suppose Achilles allows the tortoise a head start of 100 yards. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 yards, bringing him to the tortoise s starting point. During this time, the tortoise has run a much shorter distance, say, 10 yards. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. To Infinity and Beyond 4/43
Clicker Question Do you think Achilles will never catch up to the Tortoise? A Yes B No To Infinity and Beyond 5/43
A Football Paradox The 49ers and Cowboys are playing. The 49ers have the ball on the 1 yard line. As they start the play the Cowboys are called off sides. The penalty moves the ball half the distance to the goal line, so now it is on the 1/2 yard line. Again, the Cowboys are off sides; the penalty moves the ball half the distance to the goal line. They keep being called for off sides. No matter how many penalties, the ball is not quite to the goal line. If they get called for infinitely many penalties, shouldn t the 49ers end up in the end zone? To Infinity and Beyond 6/43
Clicker Question Do you think the 49ers should end up in the end zone because of offsides penalties? A Yes B No C I don t know To Infinity and Beyond 7/43
A Video about these Paradoxes We ll watch a couple minutes of a YouTube video that describes these two paradoxes. On Thursday we ll watch more of the video. Zeno s Paradox - Numberphile To Infinity and Beyond 8/43
Galileo s Paradox To Infinity and Beyond 9/43
The following comes from Galileo s book Dialogue Concerning the Two New Sciences (from books.google.com) In readable form... To Infinity and Beyond 10/43
Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension. Salviati: This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty. I take it for granted that you know which of the numbers are squares and which are not. To Infinity and Beyond 11/43
Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves. Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not? Simplicio: Most certainly. Salviati: If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. To Infinity and Beyond 12/43
Simplicio: Precisely so. Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as the numbers because every number is the root of some square. This being granted, we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said that there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers taken all together. To Infinity and Beyond 13/43
Sagredo: What then must one conclude under these circumstances? Salviati: So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes equal, greater, and less, are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. To Infinity and Beyond 14/43
To put this more briefly, the paradox is that it seems that the set of all square numbers {1, 4, 9, 16, 25,...} is both smaller than and the same size as the set of all whole numbers {1, 2, 3, 4, 5,...}. Both cannot be true. Galileo gets the idea that these two sets may be the same size by seeing that elements can be paired off, without leaving out anything. To Infinity and Beyond 15/43
To give a simpler example of this idea, If I have 10 M&Ms and you take 8 of them, then you have fewer M&Ms than I started with. Galileo s paradox makes this basic fact not clear for infinite sets. So, is the set of all square numbers smaller than the set of all whole numbers? Is it the same size? Does the question even make sense? To Infinity and Beyond 16/43
Clicker Question Do you think all infinite sets the same size? A Yes B No To Infinity and Beyond 17/43
Georg Cantor To Infinity and Beyond 18/43
Cantor, a German mathematician, whose career was mostly in the later part of the 19th century, did important work in set theory. He formalized the idea of the size of a set, and defined what it means for one set to be larger, smaller, or the same size as another set. He used the idea Galileo discussed. His work, applied to infinite sets, was harshly criticized by many, including some of the most famous mathematicians of the time. One, David Hilbert, strongly supported his work. We will revisit Hilbert in the next class. To Infinity and Beyond 19/43
How to Compare Sizes of Sets Kids can compare two sets before knowing their numbers by pairing off elements. For example, a very young child can understand that there are just as many M&Ms as cars in the following picture. To Infinity and Beyond 20/43
Roughly, two sets have the same size if one can pair off elements of one set with elements of the other, leaving no elements left out of the pairing. One set is larger than another if the second is the same size as a subset of the first, but not vice-versa. To Infinity and Beyond 21/43
Infinite Sets Cantor extended this idea to arbitrary sets by saying two sets have the same size, whether or not they are finite or infinite, if elements of one can be paired with elements of the other, leaving no elements out. With his definition, it is true that the set of whole numbers {1, 2, 3, 4, 5,...} is the same size as the set of squares {1, 4, 9, 16, 25,...}, by what Salviati says in Galileo s dialogue. Cantor also defined a set to be infinite if it is the same size as a proper subset. It is not at all obvious that this corresponds to our intuition. To Infinity and Beyond 22/43
Using Cantor s definition of infinite set, we see that the set of whole numbers is infinite, because it is the same size as the set squares. To Infinity and Beyond 23/43
Galileo s paradox arises from the thought that one should be able to say two sets are the same size if you can pair up elements, which makes perfect sense for finite sets. Galileo thought that the set of squares shouldn t be as large as the set of whole numbers, even though they can be paired off. With Cantor s definition of infinity, any infinite set would be just as mysterious to Galileo. To Infinity and Beyond 24/43
David Hilbert To Infinity and Beyond 25/43
David Hilbert was a German mathematician who lived from 1862-1943. He was considered one of the top mathematicians in the world at the beginning of the 20th century, and he influenced many areas of mathematics. While many mathematicians derided Cantor for his work on set theory, Hilbert was a strong supporter of Cantor s work. The puzzles we will present are due to Hilbert, hence the name of this lecture. To Infinity and Beyond 26/43
The Hilbert Hotel There are several puzzles that go under the title The Hibert Hotel. We will discuss several of these. The Hilbert Hotel is an unusual hotel. There is one room for every whole number. That is, there is Room 1, Room 2, and so on. It has infinitely many rooms. It would be a drag having to clean it! To Infinity and Beyond 27/43
Hilbert Hotel Question 1 It is a dark and stormy Friday the 13th. The Hilbert Hotel is fully booked. Late that night a traveller comes to the hotel hoping for a room. There is a sign outside the hotel saying Fully Booked. Confused that the sign does not say no vacancy, the traveller entered the lobby and inquires about a room. The hotel clerk says that in spite of every room being occupied, he can accommodate the traveller without having to kick anybody out of the hotel or have guests share rooms. Can he do this? To Infinity and Beyond 28/43
Clicker Question Can the hotel clerk accommodate everybody? A Yes B No C I don t know To Infinity and Beyond 29/43
Answer The hotel clerk tells the guest in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on. He then tells the traveller to take room 1. The traveller gets a room, and every current guest gets a room. In general, the guest who was in Room n moves to Room n + 1. Therefore, nobody is kicked out and nobody has to share a room. To Infinity and Beyond 30/43
Question 2 Two couples come to the Hilbert Hotel, which is fully booked. They hope to book 2 rooms. The clerk says he can accommodate their request. How can he do this without kicking anybody out of the hotel? To Infinity and Beyond 31/43
Clicker Question Can the hotel clerk accommodate everybody, when two couples come to a fully booked Hilbert Hotel? A Yes B No C I don t know To Infinity and Beyond 32/43
Answer The clerk asks the guest in Room 1 to move to Room 3, the guest in Room 2 to move to Room 4, and so on. This frees up Rooms 1 and 2, and he puts the two couples in those two rooms. Again, the travelers are accommodated without kicking anybody out of the hotel and without guests having to share rooms. In general, the person who was in Room n gets moved to Room n + 2. To Infinity and Beyond 33/43
Question 3 There are two fully booked Hilbert Hotels in town. During a lightning storm one of the hotels burns down. The manager calls the other hotel s manager and asks if he can accommodate the first hotel s guests. The second manager says sure, I can do that. How can the second manager accommodate all the guests from the first hotel without kicking any of the current guests out of the hotel? To Infinity and Beyond 34/43
Answer This is more difficult, but the manager figures out how to handle the request. He asks the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 4, and so on. In general, the guest in Room n moves to Room 2n. This frees up Rooms 1, 3, 5, 7, and so on. He then puts the first new guest in Room 1, the second in Room 3, and so on. In general, the guest who was in Room n of the first Hilbert Hotel occupies Room 2n 1 in the second hotel. Everybody gets a room and nobody has to share. To Infinity and Beyond 35/43
In fact, if a town has any (finite) number of fully booked Hilbert Hotels, all of the guests could be accommodated in just one of them without kicking anybody out or making anybody share. If you are curious about this, think first about the case of three hotels, and see if you can see a pattern for how to do this. To Infinity and Beyond 36/43
Question A large town has one Hilbert Hotel for every whole number; that is, there is a Hilbert Hotel 1, a Hilbert Hotel 2, and so on. In a freak storm all the hotels burn down. Luckily a brand new Hilbert Hotel was been built in the next town, and is now open and ready to take on customers. Can all the guests at the destroyed hotels be accommodated in the new Hilbert Hotel? To Infinity and Beyond 37/43
Clicker Question What do you think? Can one Hilbert Hotel accommodate all the guests of infinitely many Hilbert Hotels? A Yes B No To Infinity and Beyond 38/43
Answer This is not easy, but can be done! In fact, here is a way to accommodate everybody and have tons of rooms left! He assigns the person who was in Room n of hotel m room number 2 n 3 m. For example, the guest in Room 1 of hotel 1 gets Room 2 3 = 6. The guest in Room 3 of hotel 2 gets Room 2 3 3 2 = (2 2 2) (3 3) = 8 9 = 72 This allows everybody to have a room with no two guests being assigned to the same room. To Infinity and Beyond 39/43
This explanation uses a fact about whole numbers and the uniqueness of factorization into prime numbers. The numbers 2 and 3 are primes. No number can be factored into a product of primes (e.g., 2 and 3) in two different ways. For example, we can t have a product of five 2s and seven 3s equal to a product of six 2s and six 3s. This is why nobody has to share a room. This scheme leaves many rooms (infinitely many!) vacant. For example, Rooms 1, 5, 7, 10, 11, 13, 14 are but a few of those left vacant. To Infinity and Beyond 40/43
Here is a bit of a diagram to show what is happening for some of the hotels and rooms. To Infinity and Beyond 41/43
Let s watch a video from ed.ted.com about the Hilbert Hotel problem. Hilbert Hotel video from ed.ted.com To Infinity and Beyond 42/43
Next Time We will get back to the paradoxes discussed on Monday and we will also address the question, posed earlier, whether all infinite sets have the same size, according to Cantor s notion of size. We will also see some other instances of infinity and some videos about them. To Infinity and Beyond 43/43