1 00:00:00,500 --> 00:00:03,800 For those of you who are curious, we will go through an 2 00:00:03,800 --> 00:00:07,420 argument that establishes that the set of real numbers is an 3 00:00:07,420 --> 00:00:08,830 uncountable set. 4 00:00:08,830 --> 00:00:11,060 It's a famous argument known as Cantor's 5 00:00:11,060 --> 00:00:13,570 diagonalization argument. 6 00:00:13,570 --> 00:00:16,480 Actually, instead of looking at the set of all real 7 00:00:16,480 --> 00:00:22,190 numbers, we will first look at the set of all numbers, x, 8 00:00:22,190 --> 00:00:26,110 that belong to the open unit interval-- 9 00:00:26,110 --> 00:00:28,880 so numbers between 0 and 1-- 10 00:00:28,880 --> 00:00:34,230 and such that their decimal expansion involves 11 00:00:34,230 --> 00:00:36,335 only threes and fours. 12 00:00:40,980 --> 00:00:45,305 Now, the choice of three and four is somewhat arbitrary. 13 00:00:45,305 --> 00:00:46,840 It doesn't matter. 14 00:00:46,840 --> 00:00:49,700 What really matters is that we do not have 15 00:00:49,700 --> 00:00:52,280 long strings of nines. 16 00:00:52,280 --> 00:00:57,300 So suppose that this set was countable. 17 00:00:57,300 --> 00:01:04,440 If the set was countable, then that set could be written as 18 00:01:04,440 --> 00:01:11,720 equal to a set of this form, x1, x2, x3 and so on, where 19 00:01:11,720 --> 00:01:18,280 each one of these is a real number inside that set. 20 00:01:18,280 --> 00:01:20,860 Now, suppose that this is the case. 21 00:01:20,860 --> 00:01:23,690 Let us take those numbers and write them 22 00:01:23,690 --> 00:01:26,620 down in decimal notation. 23 00:01:26,620 --> 00:01:31,760 For example, one number could be this one, and 24 00:01:31,760 --> 00:01:32,920 it continues forever. 25 00:01:32,920 --> 00:01:35,330 Since we're talking about real numbers, their decimal 26 00:01:35,330 --> 00:01:38,210 expansion will go on forever. 27 00:01:38,210 --> 00:01:43,979 Suppose that the second number is of this kind, and it has 28 00:01:43,979 --> 00:01:45,495 its own decimal expansion. 29 00:01:48,150 --> 00:01:53,430 Suppose that the third number is, again, with some decimal 30 00:01:53,430 --> 00:01:55,690 expansion and so on. 31 00:01:55,690 --> 00:02:00,790 So we have assumed that our set is countable and 32 00:02:00,790 --> 00:02:03,930 therefore, the set is equal to that sequence. 33 00:02:03,930 --> 00:02:09,470 So this sequence exhausts all the numbers in that set. 34 00:02:09,470 --> 00:02:12,270 Can it do that? 35 00:02:12,270 --> 00:02:16,220 Let's construct a new number in the following fashion. 36 00:02:16,220 --> 00:02:19,579 The new number looks at this digit and 37 00:02:19,579 --> 00:02:20,840 does something different. 38 00:02:20,840 --> 00:02:24,010 Looks at this digit, the second digit of the second 39 00:02:24,010 --> 00:02:26,420 number, and does something different. 40 00:02:26,420 --> 00:02:29,840 Looks at the third digit of the third number and does 41 00:02:29,840 --> 00:02:30,980 something different. 42 00:02:30,980 --> 00:02:33,200 And we continue this way. 43 00:02:33,200 --> 00:02:36,829 This number that we have constructed here is different 44 00:02:36,829 --> 00:02:38,120 from the first number. 45 00:02:38,120 --> 00:02:39,820 They differ in the first digit. 46 00:02:39,820 --> 00:02:41,520 It's different from the second number. 47 00:02:41,520 --> 00:02:43,470 They differ in the second digit. 48 00:02:43,470 --> 00:02:46,370 It's different from the third number because it's different 49 00:02:46,370 --> 00:02:48,780 in the third digit and so on. 50 00:02:48,780 --> 00:02:55,900 So this is a number, and this number is different 51 00:02:55,900 --> 00:03:01,310 from xi for all i. 52 00:03:01,310 --> 00:03:07,510 So we have an element of this set which does not belong to 53 00:03:07,510 --> 00:03:08,800 this sequence. 54 00:03:08,800 --> 00:03:13,970 Therefore, it cannot be true that this set is equal to the 55 00:03:13,970 --> 00:03:16,980 set formed by that sequence. 56 00:03:16,980 --> 00:03:21,610 And so this is a contradiction to the initial assumption that 57 00:03:21,610 --> 00:03:25,100 this set could be written in this form, and this 58 00:03:25,100 --> 00:03:29,560 contradiction establishes that since this is not possible, 59 00:03:29,560 --> 00:03:34,520 that the set that we have here is an uncountable set. 60 00:03:34,520 --> 00:03:38,560 Now, this set is a subset of the set of real numbers. 61 00:03:38,560 --> 00:03:42,030 Since this one is uncountable, it is not hard to show that 62 00:03:42,030 --> 00:03:46,060 the set of real numbers, which is a bigger set, will also be 63 00:03:46,060 --> 00:03:47,940 uncountable. 64 00:03:47,940 --> 00:03:51,440 And so this is this particular famous argument. 65 00:03:51,440 --> 00:03:54,570 We will not need it or make any arguments of this type in 66 00:03:54,570 --> 00:03:58,340 this class, but it's so beautiful that it's worth for 67 00:03:58,340 --> 00:04:00,910 everyone to see it once in their lifetime.