full transcript

From the Ted Talk by Dennis Wildfogel: How big is infinity?

Unscramble the Blue Letters

When I was in fourth grade, my tcaeher said to us one day: "There are as many even numbers as there are numbers." "Really?", I thought. Well, yeah, there are infinitely many of both, so I suppose there are the same number of them. But even numbers are only part of the whole numbers, all the odd numbers are left over, so there's got to be more whole numbers than even numbers, right? To see what my teacher was getting at, let's first think about what it mneas for two sets to be the same size. What do I mean when I say I have the same number of fingers on my right hand as I do on left hand? Of course, I have five fingers on each, but it's actually simpler than that. I don't have to count, I only need to see that I can mctah them up, one to one. In fact, we think that some ancient people who spoke luggenaas that didn't have words for numbers greater than three used this sort of magic. For instance, if you let your sheep out of a pen to graze, you can keep track of how many went out by setting aside a stone for each one, and putting those stones back one by one when the sheep return, so you know if any are missing without really counting. As another example of matching being more fundamental than counting, if I'm speaking to a packed auditorium, where every seat is taken and no one is standing, I know that there are the same number of chairs as people in the audience, even though I don't know how many there are of either. So, what we really mean when we say that two sets are the same size is that the elements in those sets can be matched up one by one in some way. My fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double. As you can see, the bottom row contains all the even numbers, and we have a one-to-one match. That is, there are as many even numbers as there are numbers. But what still bothers us is our dsritses over the fact that even numbers seem to be only part of the whole numbers. But does this convince you that I don't have the same number of fingers on my right hand as I do on my left? Of course not. It doesn't matter if you try to match the elements in some way and it doesn't work, that doesn't convince us of anything. If you can find one way in which the elements of two sets do match up, then we say those two sets have the same number of elements. Can you make a list of all the fractions? This might be hard, there are a lot of fractions! And it's not obvious what to put first, or how to be sure all of them are on the list. Nevertheless, there is a very clever way that we can make a list of all the fractions. This was first done by Georg conatr, in the late eighteen hundreds. First, we put all the fractions into a grid. They're all there. For instance, you can find, say, 117/243, in the 117th row and 243rd column. Now we make a list out of this by sranittg at the upper left and spnieewg back and forth diagonally, skipping over any fraction, like 2/2, that represents the same number as one the we've already picked. We get a list of all the fractions, which means we've created a one-to-one match between the whole numbers and the fractions, despite the fact that we thought maybe there ought to be more fractions. OK, here's where it gets really interesting. You may know that not all real numbers — that is, not all the numbers on a number line — are frnaictos. The square root of two and pi, for instance. Any number like this is called itiroaranl. Not because it's crazy, or anything, but because the fractions are ratios of whole numbers, and so are caleld rationals; meaning the rest are non-rational, that is, irrational. Irrationals are retserepned by infinite, non-repeating decimals. So, can we make a one-to-one match between the whole numbers and the set of all the decimals, both the rationals and the iaitonrarls? That is, can we make a list of all the decimal numbers? Cantor showed that you can't. Not merely that we don't know how, but that it can't be done. Look, suppose you cliam you have made a list of all the decimals. I'm going to show you that you didn't seeuccd, by producing a decimal that is not on your list. I'll construct my decimal one place at a time. For the first decimal place of my number, I'll look at the first decimal place of your first number. If it's a one, I'll make mine a two; otherwise I'll make mine a one. For the second place of my nuembr, I'll look at the second pclae of your second number. Again, if yours is a one, I'll make mine a two, and otherwise I'll make mine a one. See how this is going? The decimal I've poudcerd can't be on your list. Why? Could it be, say, your 143rd number? No, because the 143rd place of my dceiaml is different from the 143rd place of your 143rd number. I made it that way. Your list is incomplete. It doesn't contain my decimal number. And, no matter what list you give me, I can do the same thing, and pcroude a decimal that's not on that list. So we're faced with this astounding conclusion: The decimal numbers cannot be put on a list. They represent a bigger infinity that the infinity of whole nbmerus. So, even though we're familiar with only a few irrationals, like square root of two and pi, the infinity of irrationals is actually getaerr than the infinity of fractions. Someone once said that the rtaialnos — the fractions — are like the stars in the night sky. The irrationals are like the blackness. Cantor also showed that, for any itninife set, forming a new set made of all the subsets of the original set represents a bigger ifitniny than that original set. This means that, once you have one infinity, you can always make a bigger one by maikng the set of all subtess of that first set. And then an even bigger one by making the set of all the subsets of that one. And so on. And so, there are an infinite number of infinities of different sizes. If these ideas make you uncomfortable, you are not alone. Some of the greatest mihtncamtaieas of Cantor's day were very upset with this stuff. They tried to make these different infinities irrelevant, to make mathematics work without them somehow. Cantor was even viilefid penlrlsaoy, and it got so bad for him that he suffered severe desepoirsn, and spent the last half of his life in and out of mental institutions. But eventually, his ideas won out. Today, they're coidsneerd fundamental and magnificent. All research mathematicians accpet these ideas, every college math moajr learns them, and I've elnexapid them to you in a few miuetns. Some day, perhaps, they'll be common knowledge. There's more. We just pointed out that the set of decimal numbers — that is, the real numbers — is a bigger infinity than the set of whole numbers. Cantor wondered whether there are infinities of different sezis between these two infinities. He didn't believe there were, but couldn't prove it. Cantor's conjecture became known as the continuum hypothesis. In 1900, the great mathematician David herlbit listed the cuonitunm hypothesis as the most ipnmratot unsolved problem in mathematics. The 20th ctrenuy saw a resolution of this peolrbm, but in a completely unexpected, paradigm-shattering way. In the 1920s, Kurt Gödel showed that you can never prove that the continuum hhpeisoyts is false. Then, in the 1960s, Paul J. cheon shewod that you can never prove that the continuum hypothesis is true. Taken together, these results mean that there are unanswerable questions in mathematics. A very snnnutig conclusion. Mathematics is rightly considered the pinnacle of human reasoning, but we now know that even mathematics has its limitations. Still, mcaietthmas has some truly amazing things for us to think about.

Open Cloze

When I was in fourth grade, my _______ said to us one day: "There are as many even numbers as there are numbers." "Really?", I thought. Well, yeah, there are infinitely many of both, so I suppose there are the same number of them. But even numbers are only part of the whole numbers, all the odd numbers are left over, so there's got to be more whole numbers than even numbers, right? To see what my teacher was getting at, let's first think about what it _____ for two sets to be the same size. What do I mean when I say I have the same number of fingers on my right hand as I do on left hand? Of course, I have five fingers on each, but it's actually simpler than that. I don't have to count, I only need to see that I can _____ them up, one to one. In fact, we think that some ancient people who spoke _________ that didn't have words for numbers greater than three used this sort of magic. For instance, if you let your sheep out of a pen to graze, you can keep track of how many went out by setting aside a stone for each one, and putting those stones back one by one when the sheep return, so you know if any are missing without really counting. As another example of matching being more fundamental than counting, if I'm speaking to a packed auditorium, where every seat is taken and no one is standing, I know that there are the same number of chairs as people in the audience, even though I don't know how many there are of either. So, what we really mean when we say that two sets are the same size is that the elements in those sets can be matched up one by one in some way. My fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double. As you can see, the bottom row contains all the even numbers, and we have a one-to-one match. That is, there are as many even numbers as there are numbers. But what still bothers us is our ________ over the fact that even numbers seem to be only part of the whole numbers. But does this convince you that I don't have the same number of fingers on my right hand as I do on my left? Of course not. It doesn't matter if you try to match the elements in some way and it doesn't work, that doesn't convince us of anything. If you can find one way in which the elements of two sets do match up, then we say those two sets have the same number of elements. Can you make a list of all the fractions? This might be hard, there are a lot of fractions! And it's not obvious what to put first, or how to be sure all of them are on the list. Nevertheless, there is a very clever way that we can make a list of all the fractions. This was first done by Georg ______, in the late eighteen hundreds. First, we put all the fractions into a grid. They're all there. For instance, you can find, say, 117/243, in the 117th row and 243rd column. Now we make a list out of this by ________ at the upper left and ________ back and forth diagonally, skipping over any fraction, like 2/2, that represents the same number as one the we've already picked. We get a list of all the fractions, which means we've created a one-to-one match between the whole numbers and the fractions, despite the fact that we thought maybe there ought to be more fractions. OK, here's where it gets really interesting. You may know that not all real numbers — that is, not all the numbers on a number line — are _________. The square root of two and pi, for instance. Any number like this is called __________. Not because it's crazy, or anything, but because the fractions are ratios of whole numbers, and so are ______ rationals; meaning the rest are non-rational, that is, irrational. Irrationals are ___________ by infinite, non-repeating decimals. So, can we make a one-to-one match between the whole numbers and the set of all the decimals, both the rationals and the ___________? That is, can we make a list of all the decimal numbers? Cantor showed that you can't. Not merely that we don't know how, but that it can't be done. Look, suppose you _____ you have made a list of all the decimals. I'm going to show you that you didn't _______, by producing a decimal that is not on your list. I'll construct my decimal one place at a time. For the first decimal place of my number, I'll look at the first decimal place of your first number. If it's a one, I'll make mine a two; otherwise I'll make mine a one. For the second place of my ______, I'll look at the second _____ of your second number. Again, if yours is a one, I'll make mine a two, and otherwise I'll make mine a one. See how this is going? The decimal I've ________ can't be on your list. Why? Could it be, say, your 143rd number? No, because the 143rd place of my _______ is different from the 143rd place of your 143rd number. I made it that way. Your list is incomplete. It doesn't contain my decimal number. And, no matter what list you give me, I can do the same thing, and _______ a decimal that's not on that list. So we're faced with this astounding conclusion: The decimal numbers cannot be put on a list. They represent a bigger infinity that the infinity of whole _______. So, even though we're familiar with only a few irrationals, like square root of two and pi, the infinity of irrationals is actually _______ than the infinity of fractions. Someone once said that the _________ — the fractions — are like the stars in the night sky. The irrationals are like the blackness. Cantor also showed that, for any ________ set, forming a new set made of all the subsets of the original set represents a bigger ________ than that original set. This means that, once you have one infinity, you can always make a bigger one by ______ the set of all _______ of that first set. And then an even bigger one by making the set of all the subsets of that one. And so on. And so, there are an infinite number of infinities of different sizes. If these ideas make you uncomfortable, you are not alone. Some of the greatest ______________ of Cantor's day were very upset with this stuff. They tried to make these different infinities irrelevant, to make mathematics work without them somehow. Cantor was even ________ __________, and it got so bad for him that he suffered severe __________, and spent the last half of his life in and out of mental institutions. But eventually, his ideas won out. Today, they're __________ fundamental and magnificent. All research mathematicians ______ these ideas, every college math _____ learns them, and I've _________ them to you in a few _______. Some day, perhaps, they'll be common knowledge. There's more. We just pointed out that the set of decimal numbers — that is, the real numbers — is a bigger infinity than the set of whole numbers. Cantor wondered whether there are infinities of different _____ between these two infinities. He didn't believe there were, but couldn't prove it. Cantor's conjecture became known as the continuum hypothesis. In 1900, the great mathematician David _______ listed the _________ hypothesis as the most _________ unsolved problem in mathematics. The 20th _______ saw a resolution of this _______, but in a completely unexpected, paradigm-shattering way. In the 1920s, Kurt Gödel showed that you can never prove that the continuum __________ is false. Then, in the 1960s, Paul J. _____ ______ that you can never prove that the continuum hypothesis is true. Taken together, these results mean that there are unanswerable questions in mathematics. A very ________ conclusion. Mathematics is rightly considered the pinnacle of human reasoning, but we now know that even mathematics has its limitations. Still, ___________ has some truly amazing things for us to think about.

Solution

  1. rationals
  2. cohen
  3. century
  4. cantor
  5. hilbert
  6. important
  7. means
  8. explained
  9. represented
  10. produce
  11. irrational
  12. starting
  13. irrationals
  14. minutes
  15. depression
  16. hypothesis
  17. problem
  18. stunning
  19. produced
  20. languages
  21. distress
  22. claim
  23. mathematics
  24. greater
  25. place
  26. infinity
  27. major
  28. subsets
  29. numbers
  30. called
  31. number
  32. infinite
  33. vilified
  34. personally
  35. decimal
  36. succeed
  37. showed
  38. sizes
  39. teacher
  40. mathematicians
  41. making
  42. sweeping
  43. match
  44. accept
  45. considered
  46. fractions
  47. continuum

Original Text

When I was in fourth grade, my teacher said to us one day: "There are as many even numbers as there are numbers." "Really?", I thought. Well, yeah, there are infinitely many of both, so I suppose there are the same number of them. But even numbers are only part of the whole numbers, all the odd numbers are left over, so there's got to be more whole numbers than even numbers, right? To see what my teacher was getting at, let's first think about what it means for two sets to be the same size. What do I mean when I say I have the same number of fingers on my right hand as I do on left hand? Of course, I have five fingers on each, but it's actually simpler than that. I don't have to count, I only need to see that I can match them up, one to one. In fact, we think that some ancient people who spoke languages that didn't have words for numbers greater than three used this sort of magic. For instance, if you let your sheep out of a pen to graze, you can keep track of how many went out by setting aside a stone for each one, and putting those stones back one by one when the sheep return, so you know if any are missing without really counting. As another example of matching being more fundamental than counting, if I'm speaking to a packed auditorium, where every seat is taken and no one is standing, I know that there are the same number of chairs as people in the audience, even though I don't know how many there are of either. So, what we really mean when we say that two sets are the same size is that the elements in those sets can be matched up one by one in some way. My fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double. As you can see, the bottom row contains all the even numbers, and we have a one-to-one match. That is, there are as many even numbers as there are numbers. But what still bothers us is our distress over the fact that even numbers seem to be only part of the whole numbers. But does this convince you that I don't have the same number of fingers on my right hand as I do on my left? Of course not. It doesn't matter if you try to match the elements in some way and it doesn't work, that doesn't convince us of anything. If you can find one way in which the elements of two sets do match up, then we say those two sets have the same number of elements. Can you make a list of all the fractions? This might be hard, there are a lot of fractions! And it's not obvious what to put first, or how to be sure all of them are on the list. Nevertheless, there is a very clever way that we can make a list of all the fractions. This was first done by Georg Cantor, in the late eighteen hundreds. First, we put all the fractions into a grid. They're all there. For instance, you can find, say, 117/243, in the 117th row and 243rd column. Now we make a list out of this by starting at the upper left and sweeping back and forth diagonally, skipping over any fraction, like 2/2, that represents the same number as one the we've already picked. We get a list of all the fractions, which means we've created a one-to-one match between the whole numbers and the fractions, despite the fact that we thought maybe there ought to be more fractions. OK, here's where it gets really interesting. You may know that not all real numbers — that is, not all the numbers on a number line — are fractions. The square root of two and pi, for instance. Any number like this is called irrational. Not because it's crazy, or anything, but because the fractions are ratios of whole numbers, and so are called rationals; meaning the rest are non-rational, that is, irrational. Irrationals are represented by infinite, non-repeating decimals. So, can we make a one-to-one match between the whole numbers and the set of all the decimals, both the rationals and the irrationals? That is, can we make a list of all the decimal numbers? Cantor showed that you can't. Not merely that we don't know how, but that it can't be done. Look, suppose you claim you have made a list of all the decimals. I'm going to show you that you didn't succeed, by producing a decimal that is not on your list. I'll construct my decimal one place at a time. For the first decimal place of my number, I'll look at the first decimal place of your first number. If it's a one, I'll make mine a two; otherwise I'll make mine a one. For the second place of my number, I'll look at the second place of your second number. Again, if yours is a one, I'll make mine a two, and otherwise I'll make mine a one. See how this is going? The decimal I've produced can't be on your list. Why? Could it be, say, your 143rd number? No, because the 143rd place of my decimal is different from the 143rd place of your 143rd number. I made it that way. Your list is incomplete. It doesn't contain my decimal number. And, no matter what list you give me, I can do the same thing, and produce a decimal that's not on that list. So we're faced with this astounding conclusion: The decimal numbers cannot be put on a list. They represent a bigger infinity that the infinity of whole numbers. So, even though we're familiar with only a few irrationals, like square root of two and pi, the infinity of irrationals is actually greater than the infinity of fractions. Someone once said that the rationals — the fractions — are like the stars in the night sky. The irrationals are like the blackness. Cantor also showed that, for any infinite set, forming a new set made of all the subsets of the original set represents a bigger infinity than that original set. This means that, once you have one infinity, you can always make a bigger one by making the set of all subsets of that first set. And then an even bigger one by making the set of all the subsets of that one. And so on. And so, there are an infinite number of infinities of different sizes. If these ideas make you uncomfortable, you are not alone. Some of the greatest mathematicians of Cantor's day were very upset with this stuff. They tried to make these different infinities irrelevant, to make mathematics work without them somehow. Cantor was even vilified personally, and it got so bad for him that he suffered severe depression, and spent the last half of his life in and out of mental institutions. But eventually, his ideas won out. Today, they're considered fundamental and magnificent. All research mathematicians accept these ideas, every college math major learns them, and I've explained them to you in a few minutes. Some day, perhaps, they'll be common knowledge. There's more. We just pointed out that the set of decimal numbers — that is, the real numbers — is a bigger infinity than the set of whole numbers. Cantor wondered whether there are infinities of different sizes between these two infinities. He didn't believe there were, but couldn't prove it. Cantor's conjecture became known as the continuum hypothesis. In 1900, the great mathematician David Hilbert listed the continuum hypothesis as the most important unsolved problem in mathematics. The 20th century saw a resolution of this problem, but in a completely unexpected, paradigm-shattering way. In the 1920s, Kurt Gödel showed that you can never prove that the continuum hypothesis is false. Then, in the 1960s, Paul J. Cohen showed that you can never prove that the continuum hypothesis is true. Taken together, these results mean that there are unanswerable questions in mathematics. A very stunning conclusion. Mathematics is rightly considered the pinnacle of human reasoning, but we now know that even mathematics has its limitations. Still, mathematics has some truly amazing things for us to think about.

Frequently Occurring Word Combinations

ngrams of length 2

collocation frequency
continuum hypothesis 4
bigger infinity 3
real numbers 2
square root 2
decimal place 2
decimal numbers 2
original set 2

Important Words

  1. accept
  2. amazing
  3. ancient
  4. astounding
  5. audience
  6. auditorium
  7. bad
  8. bigger
  9. blackness
  10. bothers
  11. bottom
  12. called
  13. cantor
  14. century
  15. chairs
  16. claim
  17. clever
  18. cohen
  19. college
  20. column
  21. common
  22. completely
  23. conclusion
  24. conjecture
  25. considered
  26. construct
  27. continuum
  28. convince
  29. count
  30. counting
  31. crazy
  32. created
  33. david
  34. day
  35. decimal
  36. decimals
  37. depression
  38. diagonally
  39. distress
  40. double
  41. eighteen
  42. elements
  43. eventually
  44. explained
  45. faced
  46. fact
  47. false
  48. familiar
  49. find
  50. fingers
  51. forming
  52. fourth
  53. fraction
  54. fractions
  55. fundamental
  56. georg
  57. give
  58. grade
  59. graze
  60. great
  61. greater
  62. greatest
  63. grid
  64. gödel
  65. hand
  66. hard
  67. hilbert
  68. human
  69. hundreds
  70. hypothesis
  71. ideas
  72. important
  73. incomplete
  74. infinite
  75. infinitely
  76. infinities
  77. infinity
  78. instance
  79. institutions
  80. interesting
  81. irrational
  82. irrationals
  83. irrelevant
  84. knowledge
  85. kurt
  86. laid
  87. languages
  88. late
  89. learns
  90. left
  91. life
  92. limitations
  93. line
  94. list
  95. listed
  96. lot
  97. magic
  98. magnificent
  99. major
  100. making
  101. match
  102. matched
  103. matching
  104. math
  105. mathematician
  106. mathematicians
  107. mathematics
  108. matter
  109. meaning
  110. means
  111. mental
  112. minutes
  113. missing
  114. night
  115. number
  116. numbers
  117. obvious
  118. odd
  119. original
  120. packed
  121. part
  122. paul
  123. pen
  124. people
  125. personally
  126. pi
  127. picked
  128. pinnacle
  129. place
  130. pointed
  131. problem
  132. produce
  133. produced
  134. producing
  135. prove
  136. put
  137. putting
  138. questions
  139. rationals
  140. ratios
  141. real
  142. reasoning
  143. represent
  144. represented
  145. represents
  146. research
  147. resolution
  148. rest
  149. results
  150. return
  151. rightly
  152. root
  153. row
  154. seat
  155. set
  156. sets
  157. setting
  158. severe
  159. sheep
  160. show
  161. showed
  162. simpler
  163. size
  164. sizes
  165. skipping
  166. sky
  167. sort
  168. speaking
  169. spent
  170. spoke
  171. square
  172. standing
  173. stars
  174. starting
  175. stone
  176. stones
  177. stuff
  178. stunning
  179. subsets
  180. succeed
  181. suffered
  182. suppose
  183. sweeping
  184. teacher
  185. thought
  186. time
  187. today
  188. track
  189. true
  190. unanswerable
  191. uncomfortable
  192. unexpected
  193. unsolved
  194. upper
  195. upset
  196. vilified
  197. won
  198. wondered
  199. words
  200. work
  201. yeah