[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:06.53,0:00:07.69,Default,,0000,0000,0000,,In the 1920's,
Dialogue: 0,0:00:07.72,0:00:10.18,Default,,0000,0000,0000,,the German mathematician David Hilbert
Dialogue: 0,0:00:10.21,0:00:12.44,Default,,0000,0000,0000,,devised a famous thought experiment
Dialogue: 0,0:00:12.46,0:00:14.19,Default,,0000,0000,0000,,to show us just how hard it is
Dialogue: 0,0:00:14.22,0:00:17.66,Default,,0000,0000,0000,,to wrap our minds\Naround the concept of infinity.
Dialogue: 0,0:00:18.35,0:00:21.66,Default,,0000,0000,0000,,Imagine a hotel with an infinite\Nnumber of rooms
Dialogue: 0,0:00:21.68,0:00:23.99,Default,,0000,0000,0000,,and a very hardworking night manager.
Dialogue: 0,0:00:24.53,0:00:27.52,Default,,0000,0000,0000,,One night, the Infinite Hotel\Nis completely full,
Dialogue: 0,0:00:27.55,0:00:31.00,Default,,0000,0000,0000,,totally booked up\Nwith an infinite number of guests.
Dialogue: 0,0:00:31.03,0:00:34.16,Default,,0000,0000,0000,,A man walks into the hotel\Nand asks for a room.
Dialogue: 0,0:00:34.18,0:00:35.44,Default,,0000,0000,0000,,Rather than turn him down,
Dialogue: 0,0:00:35.47,0:00:37.91,Default,,0000,0000,0000,,the night manager decides\Nto make room for him.
Dialogue: 0,0:00:37.93,0:00:38.95,Default,,0000,0000,0000,,How?
Dialogue: 0,0:00:38.97,0:00:41.64,Default,,0000,0000,0000,,Easy, he asks the guest in room number 1
Dialogue: 0,0:00:41.66,0:00:43.71,Default,,0000,0000,0000,,to move to room 2,
Dialogue: 0,0:00:43.74,0:00:46.06,Default,,0000,0000,0000,,the guest in room 2 to move to room 3,
Dialogue: 0,0:00:46.08,0:00:47.14,Default,,0000,0000,0000,,and so on.
Dialogue: 0,0:00:47.45,0:00:49.84,Default,,0000,0000,0000,,Every guest moves from room number "n"
Dialogue: 0,0:00:49.86,0:00:52.18,Default,,0000,0000,0000,,to room number "n+1".
Dialogue: 0,0:00:52.72,0:00:54.81,Default,,0000,0000,0000,,Since there are an infinite\Nnumber of rooms,
Dialogue: 0,0:00:54.84,0:00:57.01,Default,,0000,0000,0000,,there is a new room\Nfor each existing guest.
Dialogue: 0,0:00:57.41,0:00:59.76,Default,,0000,0000,0000,,This leaves room 1 open\Nfor the new customer.
Dialogue: 0,0:00:59.78,0:01:01.07,Default,,0000,0000,0000,,The process can be repeated
Dialogue: 0,0:01:01.09,0:01:03.51,Default,,0000,0000,0000,,for any finite number of new guests.
Dialogue: 0,0:01:03.54,0:01:07.53,Default,,0000,0000,0000,,If, say, a tour bus unloads\N40 new people looking for rooms,
Dialogue: 0,0:01:07.55,0:01:09.64,Default,,0000,0000,0000,,then every existing guest just moves
Dialogue: 0,0:01:09.67,0:01:10.98,Default,,0000,0000,0000,,from room number "n"
Dialogue: 0,0:01:11.00,0:01:13.64,Default,,0000,0000,0000,,to room number "n+40",
Dialogue: 0,0:01:13.66,0:01:16.20,Default,,0000,0000,0000,,thus, opening up the first 40 rooms.
Dialogue: 0,0:01:17.16,0:01:19.17,Default,,0000,0000,0000,,But now an infinitely large bus
Dialogue: 0,0:01:19.20,0:01:21.74,Default,,0000,0000,0000,,with a countably infinite\Nnumber of passengers
Dialogue: 0,0:01:21.77,0:01:23.67,Default,,0000,0000,0000,,pulls up to rent rooms.
Dialogue: 0,0:01:23.70,0:01:25.69,Default,,0000,0000,0000,,countably infinite is the key.
Dialogue: 0,0:01:26.16,0:01:28.53,Default,,0000,0000,0000,,Now, the infinite bus\Nof infinite passengers
Dialogue: 0,0:01:28.55,0:01:30.52,Default,,0000,0000,0000,,perplexes the night manager at first,
Dialogue: 0,0:01:30.54,0:01:32.01,Default,,0000,0000,0000,,but he realizes there's a way
Dialogue: 0,0:01:32.03,0:01:33.35,Default,,0000,0000,0000,,to place each new person.
Dialogue: 0,0:01:33.37,0:01:36.39,Default,,0000,0000,0000,,He asks the guest in room 1\Nto move to room 2.
Dialogue: 0,0:01:36.42,0:01:38.53,Default,,0000,0000,0000,,He then asks the guest in room 2
Dialogue: 0,0:01:38.55,0:01:40.44,Default,,0000,0000,0000,,to move to room 4,
Dialogue: 0,0:01:40.46,0:01:42.81,Default,,0000,0000,0000,,the guest in room 3 to move to room 6,
Dialogue: 0,0:01:42.83,0:01:44.10,Default,,0000,0000,0000,,and so on.
Dialogue: 0,0:01:44.13,0:01:47.31,Default,,0000,0000,0000,,Each current guest moves\Nfrom room number "n"
Dialogue: 0,0:01:47.34,0:01:49.03,Default,,0000,0000,0000,,to room number "2n" --
Dialogue: 0,0:01:50.81,0:01:54.06,Default,,0000,0000,0000,,filling up only the infinite\Neven-numbered rooms.
Dialogue: 0,0:01:54.08,0:01:55.93,Default,,0000,0000,0000,,By doing this, he has now emptied
Dialogue: 0,0:01:55.95,0:01:58.87,Default,,0000,0000,0000,,all of the infinitely many\Nodd-numbered rooms,
Dialogue: 0,0:01:58.89,0:02:02.51,Default,,0000,0000,0000,,which are then taken by the people\Nfiling off the infinite bus.
Dialogue: 0,0:02:03.24,0:02:06.88,Default,,0000,0000,0000,,Everyone's happy and the hotel's business\Nis booming more than ever.
Dialogue: 0,0:02:06.90,0:02:10.42,Default,,0000,0000,0000,,Well, actually, it is booming\Nexactly the same amount as ever,
Dialogue: 0,0:02:10.44,0:02:12.92,Default,,0000,0000,0000,,banking an infinite number\Nof dollars a night.
Dialogue: 0,0:02:14.08,0:02:16.36,Default,,0000,0000,0000,,Word spreads about this incredible hotel.
Dialogue: 0,0:02:16.38,0:02:18.54,Default,,0000,0000,0000,,People pour in from far and wide.
Dialogue: 0,0:02:18.57,0:02:20.84,Default,,0000,0000,0000,,One night, the unthinkable happens.
Dialogue: 0,0:02:20.87,0:02:23.41,Default,,0000,0000,0000,,The night manager looks outside
Dialogue: 0,0:02:23.43,0:02:27.52,Default,,0000,0000,0000,,and sees an infinite line\Nof infinitely large buses,
Dialogue: 0,0:02:27.54,0:02:30.33,Default,,0000,0000,0000,,each with a countably infinite\Nnumber of passengers.
Dialogue: 0,0:02:30.35,0:02:31.39,Default,,0000,0000,0000,,What can he do?
Dialogue: 0,0:02:31.41,0:02:34.21,Default,,0000,0000,0000,,If he cannot find rooms for them,\Nthe hotel will lose out
Dialogue: 0,0:02:34.23,0:02:35.96,Default,,0000,0000,0000,,on an infinite amount of money,
Dialogue: 0,0:02:35.98,0:02:37.96,Default,,0000,0000,0000,,and he will surely lose his job.
Dialogue: 0,0:02:37.98,0:02:41.79,Default,,0000,0000,0000,,Luckily, he remembers\Nthat around the year 300 B.C.E.,
Dialogue: 0,0:02:41.81,0:02:44.73,Default,,0000,0000,0000,,Euclid proved that there\Nis an infinite quantity
Dialogue: 0,0:02:44.75,0:02:46.63,Default,,0000,0000,0000,,of prime numbers.
Dialogue: 0,0:02:47.37,0:02:49.66,Default,,0000,0000,0000,,So, to accomplish this\Nseemingly impossible task
Dialogue: 0,0:02:49.68,0:02:52.28,Default,,0000,0000,0000,,of finding infinite beds\Nfor infinite buses
Dialogue: 0,0:02:52.31,0:02:54.29,Default,,0000,0000,0000,,of infinite weary travelers,
Dialogue: 0,0:02:54.32,0:02:57.18,Default,,0000,0000,0000,,the night manager assigns\Nevery current guest
Dialogue: 0,0:02:57.21,0:02:59.04,Default,,0000,0000,0000,,to the first prime number, 2,
Dialogue: 0,0:02:59.07,0:03:01.87,Default,,0000,0000,0000,,raised to the power\Nof their current room number.
Dialogue: 0,0:03:01.89,0:03:04.54,Default,,0000,0000,0000,,So, the current occupant of room number 7
Dialogue: 0,0:03:04.56,0:03:07.54,Default,,0000,0000,0000,,goes to room number 2^7,
Dialogue: 0,0:03:07.56,0:03:09.26,Default,,0000,0000,0000,,which is room 128.
Dialogue: 0,0:03:10.24,0:03:13.76,Default,,0000,0000,0000,,The night manager then takes the people\Non the first of the infinite buses
Dialogue: 0,0:03:13.78,0:03:15.81,Default,,0000,0000,0000,,and assigns them to the room number
Dialogue: 0,0:03:15.83,0:03:18.29,Default,,0000,0000,0000,,of the next prime, 3,
Dialogue: 0,0:03:18.32,0:03:21.73,Default,,0000,0000,0000,,raised to the power of their seat\Nnumber on the bus.
Dialogue: 0,0:03:21.75,0:03:25.26,Default,,0000,0000,0000,,So, the person in seat\Nnumber 7 on the first bus
Dialogue: 0,0:03:25.28,0:03:28.36,Default,,0000,0000,0000,,goes to room number 3^7
Dialogue: 0,0:03:28.38,0:03:31.61,Default,,0000,0000,0000,,or room number 2,187.
Dialogue: 0,0:03:31.63,0:03:34.07,Default,,0000,0000,0000,,This continues for all of the first bus.
Dialogue: 0,0:03:34.09,0:03:35.74,Default,,0000,0000,0000,,The passengers on the second bus
Dialogue: 0,0:03:35.76,0:03:39.41,Default,,0000,0000,0000,,are assigned powers of the next prime, 5.
Dialogue: 0,0:03:39.43,0:03:41.49,Default,,0000,0000,0000,,The following bus, powers of 7.
Dialogue: 0,0:03:41.52,0:03:42.92,Default,,0000,0000,0000,,Each bus follows:
Dialogue: 0,0:03:42.94,0:03:44.75,Default,,0000,0000,0000,,powers of 11, powers of 13,
Dialogue: 0,0:03:44.77,0:03:46.80,Default,,0000,0000,0000,,powers of 17, etc.
Dialogue: 0,0:03:47.37,0:03:48.73,Default,,0000,0000,0000,,Since each of these numbers
Dialogue: 0,0:03:48.75,0:03:50.97,Default,,0000,0000,0000,,only has 1 and the natural number powers
Dialogue: 0,0:03:50.99,0:03:53.21,Default,,0000,0000,0000,,of their prime number base as factors,
Dialogue: 0,0:03:53.24,0:03:55.39,Default,,0000,0000,0000,,there are no overlapping room numbers.
Dialogue: 0,0:03:55.41,0:03:58.34,Default,,0000,0000,0000,,All the buses' passengers\Nfan out into rooms
Dialogue: 0,0:03:58.36,0:04:00.85,Default,,0000,0000,0000,,using unique room-assignment schemes
Dialogue: 0,0:04:00.87,0:04:03.49,Default,,0000,0000,0000,,based on unique prime numbers.
Dialogue: 0,0:04:03.51,0:04:05.70,Default,,0000,0000,0000,,In this way, the night\Nmanager can accommodate
Dialogue: 0,0:04:05.72,0:04:07.85,Default,,0000,0000,0000,,every passenger on every bus.
Dialogue: 0,0:04:07.87,0:04:11.06,Default,,0000,0000,0000,,Although, there will be\Nmany rooms that go unfilled,
Dialogue: 0,0:04:11.09,0:04:12.36,Default,,0000,0000,0000,,like room 6,
Dialogue: 0,0:04:12.38,0:04:15.10,Default,,0000,0000,0000,,since 6 is not a power\Nof any prime number.
Dialogue: 0,0:04:15.12,0:04:17.51,Default,,0000,0000,0000,,Luckily, his bosses\Nweren't very good in math,
Dialogue: 0,0:04:17.54,0:04:18.88,Default,,0000,0000,0000,,so his job is safe.
Dialogue: 0,0:04:19.51,0:04:22.01,Default,,0000,0000,0000,,The night manager's strategies\Nare only possible
Dialogue: 0,0:04:22.03,0:04:26.62,Default,,0000,0000,0000,,because while the Infinite Hotel\Nis certainly a logistical nightmare,
Dialogue: 0,0:04:26.65,0:04:29.96,Default,,0000,0000,0000,,it only deals with the lowest\Nlevel of infinity,
Dialogue: 0,0:04:29.98,0:04:33.51,Default,,0000,0000,0000,,mainly, the countable infinity\Nof the natural numbers,
Dialogue: 0,0:04:33.54,0:04:36.59,Default,,0000,0000,0000,,1, 2, 3, 4, and so on.
Dialogue: 0,0:04:36.62,0:04:40.51,Default,,0000,0000,0000,,Georg Cantor called this level\Nof infinity aleph-zero.
Dialogue: 0,0:04:40.94,0:04:43.04,Default,,0000,0000,0000,,We use natural numbers\Nfor the room numbers
Dialogue: 0,0:04:43.07,0:04:45.19,Default,,0000,0000,0000,,as well as the seat numbers on the buses.
Dialogue: 0,0:04:45.91,0:04:48.25,Default,,0000,0000,0000,,If we were dealing\Nwith higher orders of infinity,
Dialogue: 0,0:04:48.28,0:04:49.85,Default,,0000,0000,0000,,such as that of the real numbers,
Dialogue: 0,0:04:49.87,0:04:52.84,Default,,0000,0000,0000,,these structured strategies\Nwould no longer be possible
Dialogue: 0,0:04:52.87,0:04:56.40,Default,,0000,0000,0000,,as we have no way\Nto systematically include every number.
Dialogue: 0,0:04:57.00,0:04:58.80,Default,,0000,0000,0000,,The Real Number Infinite Hotel
Dialogue: 0,0:04:58.83,0:05:00.90,Default,,0000,0000,0000,,has negative number rooms in the basement,
Dialogue: 0,0:05:00.93,0:05:02.36,Default,,0000,0000,0000,,fractional rooms,
Dialogue: 0,0:05:02.39,0:05:04.48,Default,,0000,0000,0000,,so the guy in room 1/2 always suspects
Dialogue: 0,0:05:04.51,0:05:07.18,Default,,0000,0000,0000,,he has less room than the guy in room 1.
Dialogue: 0,0:05:07.20,0:05:10.31,Default,,0000,0000,0000,,Square root rooms, like room radical 2,
Dialogue: 0,0:05:10.33,0:05:11.44,Default,,0000,0000,0000,,and room pi,
Dialogue: 0,0:05:11.46,0:05:14.32,Default,,0000,0000,0000,,where the guests expect free dessert.
Dialogue: 0,0:05:14.35,0:05:17.37,Default,,0000,0000,0000,,What self-respecting night manager\Nwould ever want to work there
Dialogue: 0,0:05:17.40,0:05:19.00,Default,,0000,0000,0000,,even for an infinite salary?
Dialogue: 0,0:05:19.03,0:05:20.92,Default,,0000,0000,0000,,But over at Hilbert's Infinite Hotel,
Dialogue: 0,0:05:20.94,0:05:22.42,Default,,0000,0000,0000,,where there's never any vacancy
Dialogue: 0,0:05:22.44,0:05:24.00,Default,,0000,0000,0000,,and always room for more,
Dialogue: 0,0:05:24.03,0:05:26.93,Default,,0000,0000,0000,,the scenarios faced by the ever-diligent
Dialogue: 0,0:05:26.95,0:05:28.76,Default,,0000,0000,0000,,and maybe too hospitable night manager
Dialogue: 0,0:05:28.78,0:05:31.49,Default,,0000,0000,0000,,serve to remind us of just how hard it is
Dialogue: 0,0:05:31.51,0:05:33.90,Default,,0000,0000,0000,,for our relatively finite minds
Dialogue: 0,0:05:33.93,0:05:36.77,Default,,0000,0000,0000,,to grasp a concept as large as infinity.
Dialogue: 0,0:05:37.13,0:05:39.08,Default,,0000,0000,0000,,Maybe you can help tackle these problems
Dialogue: 0,0:05:39.11,0:05:40.40,Default,,0000,0000,0000,,after a good night's sleep.
Dialogue: 0,0:05:40.43,0:05:42.28,Default,,0000,0000,0000,,But honestly, we might need you
Dialogue: 0,0:05:42.30,0:05:44.70,Default,,0000,0000,0000,,to change rooms at 2 a.m.