The Infinite Hotel Paradox - Jeff Dekofsky — Transcript

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1. 0:06In the 1920's,
2. 0:07the German mathematician David Hilbert
3. 0:10devised a famous thought experiment
4. 0:12to show us just how hard it is
5. 0:14to wrap our minds around the concept of infinity.
6. 0:18Imagine a hotel with an infinite number of rooms
7. 0:21and a very hardworking night manager.
8. 0:24One night, the Infinite Hotel is completely full,
9. 0:27totally booked up with an infinite number of guests.
10. 0:31A man walks into the hotel and asks for a room.
11. 0:34Rather than turn him down,
12. 0:35the night manager decides to make room for him.
13. 0:37How?
14. 0:38Easy, he asks the guest in room number 1
15. 0:41to move to room 2,
16. 0:43the guest in room 2 to move to room 3,
17. 0:46and so on.
18. 0:47Every guest moves from room number "n"
19. 0:49to room number "n+1".
20. 0:52Since there are an infinite number of rooms,
21. 0:54there is a new room for each existing guest.
22. 0:57This leaves room 1 open for the new customer.
23. 0:59The process can be repeated
24. 1:01for any finite number of new guests.
25. 1:03If, say, a tour bus unloads 40 new people looking for rooms,
26. 1:07then every existing guest just moves
27. 1:09from room number "n"
28. 1:11to room number "n+40",
29. 1:13thus, opening up the first 40 rooms.
30. 1:17But now an infinitely large bus
31. 1:19with a countably infinite number of passengers
32. 1:21pulls up to rent rooms.
33. 1:23countably infinite is the key.
34. 1:26Now, the infinite bus of infinite passengers
35. 1:28perplexes the night manager at first,
36. 1:30but he realizes there's a way
37. 1:32to place each new person.
38. 1:33He asks the guest in room 1 to move to room 2.
39. 1:36He then asks the guest in room 2
40. 1:38to move to room 4,
41. 1:40the guest in room 3 to move to room 6,
42. 1:42and so on.
43. 1:44Each current guest moves from room number "n"
44. 1:47to room number "2n" --
45. 1:50filling up only the infinite even-numbered rooms.
46. 1:54By doing this, he has now emptied
47. 1:55all of the infinitely many odd-numbered rooms,
48. 1:58which are then taken by the people filing off the infinite bus.
49. 2:03Everyone's happy and the hotel's business is booming more than ever.
50. 2:06Well, actually, it is booming exactly the same amount as ever,
51. 2:10banking an infinite number of dollars a night.
52. 2:14Word spreads about this incredible hotel.
53. 2:16People pour in from far and wide.
54. 2:18One night, the unthinkable happens.
55. 2:20The night manager looks outside
56. 2:23and sees an infinite line of infinitely large buses,
57. 2:27each with a countably infinite number of passengers.
58. 2:30What can he do?
59. 2:31If he cannot find rooms for them, the hotel will lose out
60. 2:34on an infinite amount of money,
61. 2:35and he will surely lose his job.
62. 2:37Luckily, he remembers that around the year 300 B.C.E.,
63. 2:41Euclid proved that there is an infinite quantity
64. 2:44of prime numbers.
65. 2:47So, to accomplish this seemingly impossible task
66. 2:49of finding infinite beds for infinite buses
67. 2:52of infinite weary travelers,
68. 2:54the night manager assigns every current guest
69. 2:57to the first prime number, 2,
70. 2:59raised to the power of their current room number.
71. 3:01So, the current occupant of room number 7
72. 3:04goes to room number 2^7,
73. 3:07which is room 128.
74. 3:10The night manager then takes the people on the first of the infinite buses
75. 3:13and assigns them to the room number
76. 3:15of the next prime, 3,
77. 3:18raised to the power of their seat number on the bus.
78. 3:21So, the person in seat number 7 on the first bus
79. 3:25goes to room number 3^7
80. 3:28or room number 2,187.
81. 3:31This continues for all of the first bus.
82. 3:34The passengers on the second bus
83. 3:35are assigned powers of the next prime, 5.
84. 3:39The following bus, powers of 7.
85. 3:41Each bus follows:
86. 3:42powers of 11, powers of 13,
87. 3:44powers of 17, etc.
88. 3:47Since each of these numbers
89. 3:48only has 1 and the natural number powers
90. 3:50of their prime number base as factors,
91. 3:53there are no overlapping room numbers.
92. 3:55All the buses' passengers fan out into rooms
93. 3:58using unique room-assignment schemes
94. 4:00based on unique prime numbers.
95. 4:03In this way, the night manager can accommodate
96. 4:05every passenger on every bus.
97. 4:07Although, there will be many rooms that go unfilled,
98. 4:11like room 6,
99. 4:12since 6 is not a power of any prime number.
100. 4:15Luckily, his bosses weren't very good in math,
101. 4:17so his job is safe.
102. 4:19The night manager's strategies are only possible
103. 4:22because while the Infinite Hotel is certainly a logistical nightmare,
104. 4:26it only deals with the lowest level of infinity,
105. 4:29mainly, the countable infinity of the natural numbers,
106. 4:331, 2, 3, 4, and so on.
107. 4:36Georg Cantor called this level of infinity aleph-zero.
108. 4:40We use natural numbers for the room numbers
109. 4:43as well as the seat numbers on the buses.
110. 4:45If we were dealing with higher orders of infinity,
111. 4:48such as that of the real numbers,
112. 4:49these structured strategies would no longer be possible
113. 4:52as we have no way to systematically include every number.
114. 4:57The Real Number Infinite Hotel
115. 4:58has negative number rooms in the basement,
116. 5:00fractional rooms,
117. 5:02so the guy in room 1/2 always suspects
118. 5:04he has less room than the guy in room 1.
119. 5:07Square root rooms, like room radical 2,
120. 5:10and room pi,
121. 5:11where the guests expect free dessert.
122. 5:14What self-respecting night manager would ever want to work there
123. 5:17even for an infinite salary?
124. 5:19But over at Hilbert's Infinite Hotel,
125. 5:20where there's never any vacancy
126. 5:22and always room for more,
127. 5:24the scenarios faced by the ever-diligent
128. 5:26and maybe too hospitable night manager
129. 5:28serve to remind us of just how hard it is
130. 5:31for our relatively finite minds
131. 5:33to grasp a concept as large as infinity.
132. 5:37Maybe you can help tackle these problems
133. 5:39after a good night's sleep.
134. 5:40But honestly, we might need you
135. 5:42to change rooms at 2 a.m.