1
00:00:00,500 --> 00:00:03,783
PROFESSOR: So it's time to
examine uncountable sets.

2
00:00:03,783 --> 00:00:05,866
And that's what we're going
to do in this segment.

3
00:00:10,400 --> 00:00:14,690
So Cantor's question was,
are all sets the same size?

4
00:00:14,690 --> 00:00:18,830
And he gives a
definitive answer of no.

5
00:00:18,830 --> 00:00:21,970
Cantor's theorem, which
we're about the present,

6
00:00:21,970 --> 00:00:26,460
will show that, in fact, there
isn't any biggest infinity.

7
00:00:26,460 --> 00:00:28,770
For any given infinity,
you can find a bigger one

8
00:00:28,770 --> 00:00:31,420
in a very simple way.

9
00:00:31,420 --> 00:00:35,470
But let's begin by coming
up with the simplest

10
00:00:35,470 --> 00:00:39,290
form of Cantor's
diagonal argument,

11
00:00:39,290 --> 00:00:43,470
of how do you prove that
a set is not countable?

12
00:00:43,470 --> 00:00:45,680
Remember, a set is countable
if you can list it,

13
00:00:45,680 --> 00:00:47,390
possibly with repeats.

14
00:00:47,390 --> 00:00:50,740
So A is countable if there is
a sequence, a0, a1, a2, such

15
00:00:50,740 --> 00:00:53,020
that every element
in the set A shows

16
00:00:53,020 --> 00:00:56,420
up at some time or other in the
list, possibly more than once.

17
00:00:56,420 --> 00:01:01,190
And the only things in the
list are elements of A.

18
00:01:01,190 --> 00:01:04,069
And we saw as an example
that the finite bit strings,

19
00:01:04,069 --> 00:01:06,590
the finite strings
of zeroes and ones,

20
00:01:06,590 --> 00:01:08,480
with the finite
binary words, are

21
00:01:08,480 --> 00:01:10,560
an example of a countable set.

22
00:01:10,560 --> 00:01:12,330
And we claimed last
time, and now we're

23
00:01:12,330 --> 00:01:14,890
about to prove that
the difference is

24
00:01:14,890 --> 00:01:18,680
that if you look at the infinite
bit strings-- One-way infinite.

25
00:01:18,680 --> 00:01:21,340
They have a beginning, and
they go on infinitely right.

26
00:01:21,340 --> 00:01:23,720
The notation being
0, 1 to the omega,

27
00:01:23,720 --> 00:01:26,560
where omega is an indication
of one of the symbols

28
00:01:26,560 --> 00:01:28,690
for a kind of infinity.

29
00:01:28,690 --> 00:01:32,420
And this is going to be an
example of an uncountable set.

30
00:01:32,420 --> 00:01:34,450
How are we going to prove that?

31
00:01:34,450 --> 00:01:39,630
Well, the setup for
using a diagonal argument

32
00:01:39,630 --> 00:01:41,210
is to think about
drawing a matrix.

33
00:01:41,210 --> 00:01:45,110
Suppose that I have
some way of enumerating

34
00:01:45,110 --> 00:01:50,410
the infinite binary sequences,
the 0, 1 to the omega.

35
00:01:50,410 --> 00:01:53,180
So there's a sequence s0,
there's a sequence s1,

36
00:01:53,180 --> 00:01:54,770
there's a sequence s2.

37
00:01:54,770 --> 00:01:59,350
Let's lay them out as though
they were the rows of a matrix.

38
00:01:59,350 --> 00:02:04,170
So s0 is this infinite
binary sequence, 0, 0; 1, 0;

39
00:02:04,170 --> 00:02:05,650
and so on.

40
00:02:05,650 --> 00:02:10,300
And the column labels are simply
the coordinate labels for s0.

41
00:02:10,300 --> 00:02:13,910
So this s0, 0; s0, 1; and so on.

42
00:02:13,910 --> 00:02:18,570
s1 is the next infinite
binary sequence

43
00:02:18,570 --> 00:02:20,670
in this hypothetical list.

44
00:02:20,670 --> 00:02:24,517
And it starts 0, 1, 1, 0,
and goes on, and so on,

45
00:02:24,517 --> 00:02:25,100
down the line.

46
00:02:25,100 --> 00:02:31,300
So the row labels are this
numeration of binary sequences.

47
00:02:31,300 --> 00:02:34,340
And the column labels
are coordinate labels.

48
00:02:34,340 --> 00:02:36,390
And this is a matrix that's
infinite to the right

49
00:02:36,390 --> 00:02:37,640
an infinite down.

50
00:02:37,640 --> 00:02:41,440
But it definitely has
an upper left corner.

51
00:02:41,440 --> 00:02:45,360
So the trick is to try to
find an infinite binary

52
00:02:45,360 --> 00:02:49,570
sequence that is
not in this list,

53
00:02:49,570 --> 00:02:52,270
that it differs from every row.

54
00:02:52,270 --> 00:02:54,330
If I can do that,
then I've shown

55
00:02:54,330 --> 00:02:56,990
that any attempt
to enumerate all

56
00:02:56,990 --> 00:03:01,030
of the binary sequences in 0,
1 to the omega-- any sequence

57
00:03:01,030 --> 00:03:03,740
like s0, s1, s2 of
binary sequences

58
00:03:03,740 --> 00:03:07,180
is missing something, which
means you can't really list it.

59
00:03:07,180 --> 00:03:10,910
Well, how do you find a
sequence that's not here?

60
00:03:10,910 --> 00:03:12,720
Well, pretty easy.

61
00:03:12,720 --> 00:03:14,020
You look at the first digit.

62
00:03:14,020 --> 00:03:14,960
And that was a 0.

63
00:03:14,960 --> 00:03:18,050
So you choose the first
digit of the new sequence

64
00:03:18,050 --> 00:03:19,890
to be 1, the opposite of 0.

65
00:03:19,890 --> 00:03:21,740
You choose the
second digit to be

66
00:03:21,740 --> 00:03:28,200
the opposite of the
coordinate of a digit 1 of s1,

67
00:03:28,200 --> 00:03:29,460
and complement that.

68
00:03:29,460 --> 00:03:32,630
Now here, the digit
2 of S2 is a 0.

69
00:03:32,630 --> 00:03:34,140
So let's make that a 1.

70
00:03:34,140 --> 00:03:36,990
And the next one of the
diagonal is a 0, and so on.

71
00:03:36,990 --> 00:03:40,860
We're going to complement all
of the bits along this diagonal.

72
00:03:40,860 --> 00:03:42,550
So I get a diagonal sequence.

73
00:03:42,550 --> 00:03:45,580
That's why this argument is
called the diagonal argument.

74
00:03:45,580 --> 00:03:47,830
Well, let's think about
this diagonal sequence.

75
00:03:47,830 --> 00:03:50,550
It just goes on, right down
the diagonal of this two

76
00:03:50,550 --> 00:03:52,460
dimensional infinite matrix.

77
00:03:52,460 --> 00:03:57,090
What we can say about it is
that it differs from every row.

78
00:03:57,090 --> 00:03:57,600
Why is that?

79
00:03:57,600 --> 00:04:00,450
Well, it differs from the
15th row at the 15th position,

80
00:04:00,450 --> 00:04:01,600
or coordinate 15.

81
00:04:01,600 --> 00:04:05,120
It differs from the 99th
row at coordinate 99.

82
00:04:05,120 --> 00:04:07,260
It's not in the matrix.

83
00:04:07,260 --> 00:04:09,220
It's not a row of any matrix.

84
00:04:09,220 --> 00:04:12,030
And that immediately
tells you it's over.

85
00:04:12,030 --> 00:04:16,209
Any attempt to list all of the
elements in 0, 1 to the omega

86
00:04:16,209 --> 00:04:18,490
is going to omit a
diagonal element.

87
00:04:18,490 --> 00:04:21,440
It's not possible
to list all of them.

88
00:04:21,440 --> 00:04:26,260
In other words, there
isn't any surjection

89
00:04:26,260 --> 00:04:29,870
from the non-negative
integers to 0, 1 to the omega,

90
00:04:29,870 --> 00:04:32,000
because I've just shown
you how, if you give me

91
00:04:32,000 --> 00:04:38,480
a surjection onto binary
sequences, in effect,

92
00:04:38,480 --> 00:04:42,730
I'm giving you with N a 0
sequence, a first sequence,

93
00:04:42,730 --> 00:04:44,062
a second sequence, and so on.

94
00:04:44,062 --> 00:04:46,520
Then I know exactly how to find
something that's not there.

95
00:04:46,520 --> 00:04:50,430
There can't be a surjection
from N to 0, 1 to the omega.

96
00:04:50,430 --> 00:04:52,650
It's just not true.

97
00:04:52,650 --> 00:04:55,440
And that's why we can say
that 0, 1 to the omega

98
00:04:55,440 --> 00:04:56,770
is uncountable.

99
00:04:56,770 --> 00:04:59,020
Definition of countable--
or an equivalent formulation

100
00:04:59,020 --> 00:05:00,561
of countable,
remember, is that there

101
00:05:00,561 --> 00:05:02,500
is a surjection from the
non-negative integers

102
00:05:02,500 --> 00:05:03,000
to the set.

103
00:05:03,000 --> 00:05:03,916
Well, there isn't any.

104
00:05:03,916 --> 00:05:05,310
We just proved it.

105
00:05:05,310 --> 00:05:08,110
So N is not surj 0, 1.

106
00:05:08,110 --> 00:05:11,400
By the way, it's also
quite easy to say

107
00:05:11,400 --> 00:05:14,540
that there is a surjection from
the infinite binary sequence

108
00:05:14,540 --> 00:05:16,610
to the non-negative integers.

109
00:05:16,610 --> 00:05:21,060
You could map a binary
sequence to the coordinate

110
00:05:21,060 --> 00:05:23,330
of the first one in it.

111
00:05:23,330 --> 00:05:26,210
And that maps every
infinite binary sequence

112
00:05:26,210 --> 00:05:27,500
to a non-negative integer.

113
00:05:27,500 --> 00:05:31,890
It hits every non-negative
integer lots of times.

114
00:05:31,890 --> 00:05:34,090
I hit five with a
sequence that starts

115
00:05:34,090 --> 00:05:35,540
with four zeros and a one.

116
00:05:35,540 --> 00:05:37,290
The only sequence that
doesn't go anywhere

117
00:05:37,290 --> 00:05:38,880
is the all-zero sequence.

118
00:05:38,880 --> 00:05:41,170
But by the way, if you check
the definition of surj,

119
00:05:41,170 --> 00:05:43,660
it doesn't require that
the function be total.

120
00:05:43,660 --> 00:05:47,740
Surj means that there
is a function that

121
00:05:47,740 --> 00:05:51,450
is a surjection to
N. But there doesn't

122
00:05:51,450 --> 00:05:53,954
have to be greater or
equal one arrow out.

123
00:05:53,954 --> 00:05:56,120
There just has to be less
than or equal to one arrow

124
00:05:56,120 --> 00:05:57,220
out, so it's a function.

125
00:05:57,220 --> 00:05:58,970
But of course, it's
easy enough to make it

126
00:05:58,970 --> 00:06:00,260
total, the all-zero sequence.

127
00:06:00,260 --> 00:06:01,980
Just map it to 0.

128
00:06:01,980 --> 00:06:06,850
So now you have both
the 0 and the sequence

129
00:06:06,850 --> 00:06:09,460
that starts-- Any sequence
that starts with 1

130
00:06:09,460 --> 00:06:11,100
will all map to 0.

131
00:06:11,100 --> 00:06:11,820
OK.

132
00:06:11,820 --> 00:06:15,900
So if we remember
our intuition, surj

133
00:06:15,900 --> 00:06:18,790
is read as greater
than or equal to.

134
00:06:18,790 --> 00:06:22,060
So this tells us that the
infinite binary sequences

135
00:06:22,060 --> 00:06:23,940
are a larger set, at
least as large a set

136
00:06:23,940 --> 00:06:25,210
as the non-negative integers.

137
00:06:25,210 --> 00:06:26,440
And the converse is true.

138
00:06:26,440 --> 00:06:29,890
The non-negative integers
are not at least as large

139
00:06:29,890 --> 00:06:31,650
as the infinite
binary sequences.

140
00:06:31,650 --> 00:06:34,160
So we can really say that
non-negative integers

141
00:06:34,160 --> 00:06:38,370
are strictly smaller than
the set of infinite binary

142
00:06:38,370 --> 00:06:38,950
sequences.

143
00:06:38,950 --> 00:06:41,460
Now, strictly
smaller is in quotes

144
00:06:41,460 --> 00:06:44,040
because, again, we
don't know exactly what

145
00:06:44,040 --> 00:06:46,630
the size of infinite sets is.

146
00:06:46,630 --> 00:06:48,770
All we're doing we,
really, is talking

147
00:06:48,770 --> 00:06:53,020
about properties objections,
bijections, surjections,

148
00:06:53,020 --> 00:06:54,210
injections.

149
00:06:54,210 --> 00:06:54,710
OK.

150
00:06:54,710 --> 00:06:57,620
So let's make an
explicit definition.

151
00:06:57,620 --> 00:07:01,000
I'm going to say that A
strict B means that there

152
00:07:01,000 --> 00:07:05,930
is no surjection from A to
B. So if we read A surjection

153
00:07:05,930 --> 00:07:08,960
B intuitively, as A
greater than or equal to B,

154
00:07:08,960 --> 00:07:10,980
this is saying it's
not true that A

155
00:07:10,980 --> 00:07:14,550
is greater than or equal to
B. Or in ordinary language

156
00:07:14,550 --> 00:07:16,340
and thinking about
sets, if it's not true

157
00:07:16,340 --> 00:07:17,965
that you're greater
than or equal to B,

158
00:07:17,965 --> 00:07:20,550
you must be strictly
less than B.

159
00:07:20,550 --> 00:07:22,840
So that's the motivation
for the word strict.

160
00:07:22,840 --> 00:07:25,150
But remember, we're talking
about infinite sets.

161
00:07:25,150 --> 00:07:28,870
And we can't go around assuming
too many properties of strict

162
00:07:28,870 --> 00:07:30,140
until we've proved them.

163
00:07:30,140 --> 00:07:32,140
One non-trivial
property, by the way,

164
00:07:32,140 --> 00:07:34,860
is I've defined strict
that it's not true

165
00:07:34,860 --> 00:07:37,300
that there is a
surjection from A to B,

166
00:07:37,300 --> 00:07:39,400
but I'm not insisting
that there must

167
00:07:39,400 --> 00:07:43,060
be a surjection from
B to A, which would

168
00:07:43,060 --> 00:07:44,660
be the second companion part.

169
00:07:44,660 --> 00:07:46,720
That is, A is not greater
than or equal to B,

170
00:07:46,720 --> 00:07:50,199
and B is greater than or equal
to A. Turns out, technically,

171
00:07:50,199 --> 00:07:50,990
you can prove that.

172
00:07:50,990 --> 00:07:53,710
If there isn't any
surjection from A to B,

173
00:07:53,710 --> 00:07:55,700
there will be a
surjection from B to A.

174
00:07:55,700 --> 00:07:59,130
But that's using a set theoretic
argument that's not so obvious.

175
00:07:59,130 --> 00:08:00,060
And we don't need it.

176
00:08:00,060 --> 00:08:02,090
So this is the
definition of strict.

177
00:08:02,090 --> 00:08:06,280
A straight B means you cannot
get a surjection from A to B.

178
00:08:06,280 --> 00:08:09,900
And we're intuitively reading
it as A is strictly smaller than

179
00:08:09,900 --> 00:08:13,440
B. And what we've just shown
then is that the non-negative

180
00:08:13,440 --> 00:08:17,510
integers strict
0, 1 to the omega,

181
00:08:17,510 --> 00:08:19,500
the infinite binary sequences.

182
00:08:19,500 --> 00:08:23,980
OK, now Cantor's theorem is
a wonderful generalization

183
00:08:23,980 --> 00:08:24,480
of this.

184
00:08:24,480 --> 00:08:26,979
It's a powerful generalization,
but the proof is pretty much

185
00:08:26,979 --> 00:08:29,830
the same, although it sometimes
looks a little different

186
00:08:29,830 --> 00:08:30,990
as it's written up.

187
00:08:30,990 --> 00:08:33,310
And what Cantor's
theorem says, it's

188
00:08:33,310 --> 00:08:35,650
just beautifully
elegant and simple.

189
00:08:35,650 --> 00:08:38,809
It says simply that the
power set is strictly

190
00:08:38,809 --> 00:08:40,179
better than the set.

191
00:08:40,179 --> 00:08:43,409
A strict power set of
A, for every set A,

192
00:08:43,409 --> 00:08:44,800
even if A is finite.

193
00:08:44,800 --> 00:08:48,600
Because remember, if A is
finite, say A has n elements,

194
00:08:48,600 --> 00:08:51,600
then the power set of
A has 2 the n elements.

195
00:08:51,600 --> 00:08:54,310
And you could check that
even for n equals 0,

196
00:08:54,310 --> 00:08:59,710
n is less than or equal to
the-- n is less than 2 to the n.

197
00:08:59,710 --> 00:09:03,880
0 is less than 2 to
the 0, which is 1.

198
00:09:03,880 --> 00:09:08,420
2 is less than 2 squared,
which is 4, and so on.

199
00:09:08,420 --> 00:09:11,810
So even for finite sets, we
have A strict power set of A.

200
00:09:11,810 --> 00:09:15,640
But the cool thing is that it
works even for infinite sets.

201
00:09:15,640 --> 00:09:16,620
Let's take a look.

202
00:09:16,620 --> 00:09:18,380
It's a diagonal argument again.

203
00:09:18,380 --> 00:09:21,052
But now, I mustn't assume
that A is countable.

204
00:09:21,052 --> 00:09:22,760
I'm not going to assume
that I can really

205
00:09:22,760 --> 00:09:25,020
list the elements of A.
But we'll think about it

206
00:09:25,020 --> 00:09:25,910
as though we could.

207
00:09:25,910 --> 00:09:28,080
Let's think about
this matrix again.

208
00:09:28,080 --> 00:09:31,950
So suppose A is this set of
elements, a, b, s, t, d, e.

209
00:09:31,950 --> 00:09:34,921
I'm scrambling up the
alphabet on purpose,

210
00:09:34,921 --> 00:09:37,420
because I don't want you to get
the idea that we're assuming

211
00:09:37,420 --> 00:09:40,430
that A is countable, that you
can list all the elements of A.

212
00:09:40,430 --> 00:09:41,480
I'm not assuming that.

213
00:09:41,480 --> 00:09:45,216
But I'm just writing out
a sample of elements of A.

214
00:09:45,216 --> 00:09:47,930
And let's suppose
that I was trying

215
00:09:47,930 --> 00:09:50,805
to get a surjection from
A to the power set of A.

216
00:09:50,805 --> 00:09:54,350
So suppose I have a
function f that maps each

217
00:09:54,350 --> 00:09:59,396
of the successive elements of A
to some subset of A. So f of a

218
00:09:59,396 --> 00:10:01,280
is part of the power set.

219
00:10:01,280 --> 00:10:05,345
It's a subset of capital A. f
of b is a subset of capital A,

220
00:10:05,345 --> 00:10:06,510
and so on.

221
00:10:06,510 --> 00:10:08,360
And suppose I had
a setup like this.

222
00:10:08,360 --> 00:10:11,660
I'm going to draw a matrix that
looks like the diagonal matrix.

223
00:10:11,660 --> 00:10:15,930
And we're going to extract a
diagonal set and discover that

224
00:10:15,930 --> 00:10:19,240
that diagonal set is
not one of the f's.

225
00:10:19,240 --> 00:10:21,900
It's not f of anything,
which means that f is not

226
00:10:21,900 --> 00:10:23,170
going to be a surjection.

227
00:10:23,170 --> 00:10:24,360
So let's look at it again.

228
00:10:24,360 --> 00:10:26,940
So here's this matrix,
where I'm labeling

229
00:10:26,940 --> 00:10:31,060
the columns of the matrix
by the elements of A.

230
00:10:31,060 --> 00:10:33,570
No particular order here, but
in order to draw a matrix,

231
00:10:33,570 --> 00:10:35,280
I have to write them
down in some order.

232
00:10:35,280 --> 00:10:37,460
And likewise, the
first row is going

233
00:10:37,460 --> 00:10:39,850
to be f of this element a.

234
00:10:39,850 --> 00:10:41,900
Well, what is in
f of A? f of A is

235
00:10:41,900 --> 00:10:43,520
going to be a set of elements.

236
00:10:43,520 --> 00:10:45,520
So let's just write
the elements in f

237
00:10:45,520 --> 00:10:49,260
of A down, under the
corresponding column label.

238
00:10:49,260 --> 00:10:54,190
So here's an example, where
f of A has an a and no b,

239
00:10:54,190 --> 00:10:58,090
and has it has an s
in it, no c, no d.

240
00:10:58,090 --> 00:10:59,000
It has an e.

241
00:10:59,000 --> 00:11:02,160
Likewise, f of b has an
a and b, and no s or t,

242
00:11:02,160 --> 00:11:03,330
but it's got a c, and so on.

243
00:11:03,330 --> 00:11:06,160
So I filled in this
matrix by taking

244
00:11:06,160 --> 00:11:09,955
f of an element in A, which is
supposed to be a subset of A,

245
00:11:09,955 --> 00:11:15,810
and writing out all of the
elements in that subset,

246
00:11:15,810 --> 00:11:23,390
under the corresponding
element of the subset.

247
00:11:23,390 --> 00:11:27,250
So a b goes under a b if it's
in f of c, and s goes under an s

248
00:11:27,250 --> 00:11:28,400
if it's in f of c.

249
00:11:28,400 --> 00:11:31,294
Nothing goes under a t
if t is not in f of C.

250
00:11:31,294 --> 00:11:32,710
And that's what
we're seeing here.

251
00:11:32,710 --> 00:11:34,810
So I'm laying out
as though I was

252
00:11:34,810 --> 00:11:37,190
using 0's and 1's for an
infinite binary sequence.

253
00:11:37,190 --> 00:11:40,570
I'm laying out each
of the sets that

254
00:11:40,570 --> 00:11:44,140
are in the range of
f along this row.

255
00:11:44,140 --> 00:11:48,040
And now with this setup,
and I can define a new set,

256
00:11:48,040 --> 00:11:50,160
which is not going to be an f.

257
00:11:50,160 --> 00:11:51,130
How do I get that?

258
00:11:51,130 --> 00:11:55,680
Well, what I'm going
to do is in my new set

259
00:11:55,680 --> 00:11:58,030
I'm not going to have
any of the elements that

260
00:11:58,030 --> 00:11:59,380
appear on the diagonal.

261
00:11:59,380 --> 00:12:02,520
So if a is a member
of f of A, that

262
00:12:02,520 --> 00:12:04,350
means that a appears
in this coordinate,

263
00:12:04,350 --> 00:12:06,140
it's not going to be in my set.

264
00:12:06,140 --> 00:12:11,070
If b is in f of b,
meaning that b appears

265
00:12:11,070 --> 00:12:13,000
in the f of b row
under the column b,

266
00:12:13,000 --> 00:12:15,420
it's not going to be in my set.

267
00:12:15,420 --> 00:12:19,755
On the other hand,
s is not in f of s,

268
00:12:19,755 --> 00:12:20,880
because there's no s there.

269
00:12:20,880 --> 00:12:23,940
So I'm going to put
an s there in magenta.

270
00:12:23,940 --> 00:12:26,710
And likewise, I'm
going to stick elements

271
00:12:26,710 --> 00:12:29,330
in or out the opposite
of whether they

272
00:12:29,330 --> 00:12:30,800
appear on the diagonal.

273
00:12:30,800 --> 00:12:34,330
And this is going to
give me a set D, which

274
00:12:34,330 --> 00:12:38,530
is going to be my diagonal set.

275
00:12:38,530 --> 00:12:41,150
So if we write this
out, what we're

276
00:12:41,150 --> 00:12:44,120
saying is suppose that
I have a function f

277
00:12:44,120 --> 00:12:48,700
from A to the power set of
A. Then what I'm going to do

278
00:12:48,700 --> 00:12:52,670
is define a subset of A
that's not in the range of f,

279
00:12:52,670 --> 00:12:59,390
namely set D, which is
the set of those elements

280
00:12:59,390 --> 00:13:03,670
in A, such that little
a is not an f of a.

281
00:13:03,670 --> 00:13:08,190
Namely, if an element
appeared on the diagonal

282
00:13:08,190 --> 00:13:16,200
because an element with column
label A was in the row f of a,

283
00:13:16,200 --> 00:13:18,250
then I left that out of my set.

284
00:13:18,250 --> 00:13:22,960
And if it was not in that
location in the matrix,

285
00:13:22,960 --> 00:13:24,400
I put it in my set.

286
00:13:24,400 --> 00:13:26,400
So I'm keeping all
the elements that

287
00:13:26,400 --> 00:13:30,880
aren't on the diagonal,
that's my diagonal set D.

288
00:13:30,880 --> 00:13:34,680
And what I know about it is
that D is not in the range of f,

289
00:13:34,680 --> 00:13:39,970
because it differs from every
possible row of the matrix.

290
00:13:39,970 --> 00:13:45,100
If the row is labeled
with f of a as f of a,

291
00:13:45,100 --> 00:13:50,280
then it differs in the column
a, f of a from that row.

292
00:13:50,280 --> 00:13:56,570
And therefore, my set D is
not a row of this matrix.

293
00:13:56,570 --> 00:14:00,110
And that means that it's
not equal to f of anything.

294
00:14:00,110 --> 00:14:04,580
So I've just found that there
is no f arrow into D. D's not

295
00:14:04,580 --> 00:14:05,660
in the range of f.

296
00:14:05,660 --> 00:14:09,650
That means that if I had such an
f from A to the power set of A,

297
00:14:09,650 --> 00:14:13,470
it's not a surjection,
because D is always left out.

298
00:14:13,470 --> 00:14:15,130
So f is not a surjection.

299
00:14:15,130 --> 00:14:18,540
And since, you know, f
is any function from A

300
00:14:18,540 --> 00:14:21,340
to the power set of A, none
of them are surjections,

301
00:14:21,340 --> 00:14:24,240
and that means there is no
surjection from A to the power

302
00:14:24,240 --> 00:14:28,240
set of A. In other words,
A strict power set of A.

303
00:14:28,240 --> 00:14:30,580
There is no surjection.

304
00:14:30,580 --> 00:14:33,750
Now, a special case
of this, of course,

305
00:14:33,750 --> 00:14:38,710
is that the non-negative
integers are strictly

306
00:14:38,710 --> 00:14:40,740
smaller than the power
set of N. Of course,

307
00:14:40,740 --> 00:14:43,920
that's just an instance
of Cantor's theorem.

308
00:14:43,920 --> 00:14:48,087
We're applying A being the
set of non-negative integers.

309
00:14:48,087 --> 00:14:50,420
So there is no surjection
from the non-negative integers

310
00:14:50,420 --> 00:14:53,520
to the subsets of
non-negative integers.

311
00:14:53,520 --> 00:14:55,390
Again, that means that
the power set of N

312
00:14:55,390 --> 00:14:57,730
is an example of
an uncountable set,

313
00:14:57,730 --> 00:15:00,420
because the definition
of countable

314
00:15:00,420 --> 00:15:03,050
is that there would
be a surjection from N

315
00:15:03,050 --> 00:15:05,190
to power set of N. We're
saying there isn't any,

316
00:15:05,190 --> 00:15:06,790
so it's not countable.

317
00:15:06,790 --> 00:15:09,900
Not countable is usually
phrased as uncountable.

318
00:15:09,900 --> 00:15:13,630
So the power set of N is
maybe our second example

319
00:15:13,630 --> 00:15:15,650
of an uncountable
set, the first one

320
00:15:15,650 --> 00:15:19,330
being the infinite
sequences of binary numbers.

321
00:15:19,330 --> 00:15:23,810
Now as a matter of fact,
just as we had a general way

322
00:15:23,810 --> 00:15:27,720
to prove countability-- you can
show that a set is countable

323
00:15:27,720 --> 00:15:29,773
if there is a surjection
from a set you

324
00:15:29,773 --> 00:15:31,520
know is countable
onto the target,

325
00:15:31,520 --> 00:15:32,720
then the target's countable.

326
00:15:32,720 --> 00:15:35,030
Take the contrapositive
of that lemma,

327
00:15:35,030 --> 00:15:38,940
and you can say that if
a set A is uncountable

328
00:15:38,940 --> 00:15:41,770
and there is a
surjection from C to A,

329
00:15:41,770 --> 00:15:44,120
then C has to be uncountable.

330
00:15:44,120 --> 00:15:46,270
That's just the contrapositive
of the previous one.

331
00:15:46,270 --> 00:15:48,260
If C was countable, then
A would be countable.

332
00:15:48,260 --> 00:15:51,310
So if A is uncountable,
C must be uncountable.

333
00:15:51,310 --> 00:15:53,210
So this gives us, again,
a nice general way

334
00:15:53,210 --> 00:15:55,310
to prove uncountability
of sets, once I

335
00:15:55,310 --> 00:15:57,990
have a couple in my repertoire.

336
00:15:57,990 --> 00:16:00,790
Well, it means that
we could have deduced

337
00:16:00,790 --> 00:16:04,500
that 0, 1 to the omega, that the
infinite binary sequences were

338
00:16:04,500 --> 00:16:06,560
uncountable, because
we know that there's is

339
00:16:06,560 --> 00:16:10,202
a bijection between the
infinite binary sequences

340
00:16:10,202 --> 00:16:11,160
and the power set of N.

341
00:16:11,160 --> 00:16:13,690
We described that bijection
without knowing anything

342
00:16:13,690 --> 00:16:16,984
about any other properties of
the infinite binary sequences

343
00:16:16,984 --> 00:16:19,400
in the power set of N, whether
they were countable or not.

344
00:16:19,400 --> 00:16:21,320
But now that Cantor's
theorem tells us

345
00:16:21,320 --> 00:16:23,420
that the power set
of N is uncountable

346
00:16:23,420 --> 00:16:26,470
and there's a bijection,
the previous lemma says,

347
00:16:26,470 --> 00:16:30,620
in particular, there's a
surjection from 0, 1 to omega

348
00:16:30,620 --> 00:16:33,220
to the power set of N, which
means 0, 1 to the omega

349
00:16:33,220 --> 00:16:34,290
is uncountable.

350
00:16:34,290 --> 00:16:38,820
So what I'm illustrating then
is that the proof that we used

351
00:16:38,820 --> 00:16:41,410
directly by a diagonal
argument to figure out

352
00:16:41,410 --> 00:16:45,540
that 0, 1 to the
omega was uncountable,

353
00:16:45,540 --> 00:16:48,210
it's really a special
case of the more general

354
00:16:48,210 --> 00:16:51,230
diagonal argument that we used
to prove Cantor's theorem.

355
00:16:51,230 --> 00:16:53,640
And we get that
0, 1 to the omega

356
00:16:53,640 --> 00:16:56,380
is uncountable as a
consequence of Cantor's theorem

357
00:16:56,380 --> 00:17:01,830
about the power set of N. And
so we've got two different ways

358
00:17:01,830 --> 00:17:05,010
then to prove that the
infinite binary sequences are

359
00:17:05,010 --> 00:17:07,444
uncountable.

360
00:17:07,444 --> 00:17:08,819
Another example
of an uncountable

361
00:17:08,819 --> 00:17:10,609
set, it's the real numbers.

362
00:17:10,609 --> 00:17:11,930
And they're a cute example.

363
00:17:11,930 --> 00:17:13,730
Remember, we saw that
the rational numbers

364
00:17:13,730 --> 00:17:14,665
were countable.

365
00:17:14,665 --> 00:17:16,039
The real numbers
are uncountable.

366
00:17:16,039 --> 00:17:17,440
Well, how do I prove that?

367
00:17:17,440 --> 00:17:19,079
I'm just going to
show you a surjection

368
00:17:19,079 --> 00:17:22,907
from the real numbers onto
the infinite binary sequences.

369
00:17:22,907 --> 00:17:23,990
How am I going to do that?

370
00:17:23,990 --> 00:17:27,569
Well, it's a kind of
stupid trick, but it works.

371
00:17:27,569 --> 00:17:30,190
I'm using both positive
and negative reals.

372
00:17:30,190 --> 00:17:33,050
So let's look at
some real number,

373
00:17:33,050 --> 00:17:34,780
and look at its
binary representation,

374
00:17:34,780 --> 00:17:39,440
so for the moment
that it's positive.

375
00:17:39,440 --> 00:17:42,300
So let's look at, say,
the binary representation

376
00:17:42,300 --> 00:17:43,850
of some number, like 3 and 1/3.

377
00:17:43,850 --> 00:17:45,670
So that means that
if we're thinking

378
00:17:45,670 --> 00:17:48,840
of these as binary places,
this is the 0's place,

379
00:17:48,840 --> 00:17:50,310
the 2's place, the 4's place.

380
00:17:50,310 --> 00:17:52,420
This is half's place,
the quarter's place,

381
00:17:52,420 --> 00:17:53,490
the eighth's place.

382
00:17:53,490 --> 00:17:58,360
Then the binary representation
of 3 and 1/3 would be 2.

383
00:17:58,360 --> 00:18:02,370
And then this infinite
repeating-- not decimal,

384
00:18:02,370 --> 00:18:08,220
but [? bicipal-- ?]
binary expansion, 010101.

385
00:18:08,220 --> 00:18:11,230
And we will examine how I
know that that's a third.

386
00:18:11,230 --> 00:18:12,890
But take it for
granted that that's

387
00:18:12,890 --> 00:18:14,550
what you get as the
repeated fraction.

388
00:18:14,550 --> 00:18:17,270
You could figure that out
by just doing division of 1

389
00:18:17,270 --> 00:18:19,520
by 3 in binary.

390
00:18:19,520 --> 00:18:23,200
Anyway, just as there's
a decimal expansion

391
00:18:23,200 --> 00:18:27,710
of every real number, there's
a binary expansion just using

392
00:18:27,710 --> 00:18:28,570
base 2.

393
00:18:28,570 --> 00:18:30,801
So here's the binary
expansion of 3 and 1/3.

394
00:18:30,801 --> 00:18:32,300
So what I'm going
to do is I'm going

395
00:18:32,300 --> 00:18:34,847
to map 3 and 1/3 to
this binary sequence.

396
00:18:34,847 --> 00:18:36,430
I'm going to ignore
the decimal place.

397
00:18:36,430 --> 00:18:37,970
Binary is not decimal place.

398
00:18:37,970 --> 00:18:41,360
It's a [? becimal ?]
place, or binary position.

399
00:18:41,360 --> 00:18:43,440
And I'm just going to
take this to mapping

400
00:18:43,440 --> 00:18:48,830
the sequence, 11010101.

401
00:18:48,830 --> 00:18:50,550
And I claim that
this is a surjection

402
00:18:50,550 --> 00:18:54,820
because you're going to
hit every possible binary

403
00:18:54,820 --> 00:18:56,040
sequence in this way.

404
00:18:56,040 --> 00:18:57,820
Well, almost.

405
00:18:57,820 --> 00:18:59,384
Let's take a closer look.

406
00:19:02,650 --> 00:19:09,240
There's a problem with mapping
to things that start with 0,

407
00:19:09,240 --> 00:19:14,180
because let's examine
that a half is 0.10000000.

408
00:19:14,180 --> 00:19:15,920
So I would map it to that.

409
00:19:15,920 --> 00:19:19,420
And it will end.

410
00:19:19,420 --> 00:19:22,900
But there's an ambiguity,
because a half is also equal

411
00:19:22,900 --> 00:19:32,190
to 0.011111, just as 0.999999
is equal to 1.000000 in decimal,

412
00:19:32,190 --> 00:19:35,170
you get the same infinite
carry issue here in binary.

413
00:19:35,170 --> 00:19:40,670
So numbers that end in
all ones have another way

414
00:19:40,670 --> 00:19:43,760
to represent the very same
number by a sequence that

415
00:19:43,760 --> 00:19:45,180
ends in all zeroes.

416
00:19:45,180 --> 00:19:47,070
So how am I going
to hit-- if I'm

417
00:19:47,070 --> 00:19:50,200
using up a half to hit this one,
what's left to hit that one?

418
00:19:50,200 --> 00:19:51,870
Oh, how about using minus 1/2?

419
00:19:51,870 --> 00:19:54,010
It's there, and
that's part of R,

420
00:19:54,010 --> 00:19:56,730
so I'm just going to
map the negative numbers

421
00:19:56,730 --> 00:20:00,300
to the version of the
expansion that starts with 0

422
00:20:00,300 --> 00:20:02,230
and has an infinite
number of ones.

423
00:20:02,230 --> 00:20:05,920
And the positive one that
ends with an infinite number

424
00:20:05,920 --> 00:20:06,544
of zeroes.

425
00:20:06,544 --> 00:20:08,460
And otherwise, I'm going
to map plus and minus

426
00:20:08,460 --> 00:20:10,460
numbers to the same place.

427
00:20:10,460 --> 00:20:14,220
So this is going to give me
the needed surjection from R

428
00:20:14,220 --> 00:20:16,470
to the infinite
binary sequences.

429
00:20:16,470 --> 00:20:19,600
And by our previous lemma,
that implies, sure enough,

430
00:20:19,600 --> 00:20:22,620
that the real numbers
are uncountable.