Uncountable Sets
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Countably Infinite
There are as many natural numbers as integers
0 1 2 3 4 5 6 7 8…
0, -1, 1, -2, 2, -3, 3, -4, 4 …
f(n) = n/2 if n is even, -(n+1)/2 if n is odd
is a bijection from Natural Numbers → Integers
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Infinite Sizes
Are all infinite sets the same size?
NO!
Cantor’s Theorem
shows how to keep finding
bigger infinities.
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P(N)
• How many sets of natural numbers?
• The same as there are natural numbers?
• Or more?
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Countably Infinite Sets
::= {finite bit strings}
… is countably infinite
Proof: List strings shortest to
longest, and alphabetically within
strings of the same length
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Countably infinite Sets
= {e, 0, 1, 00, 01, 10, 11, …}
= { e,
= { f(0),
0, 1,
f(1), f(2),
00, 01, 10, 11,
f(3), f(4), …}
000, … }
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Uncountably Infinite Sets
What about infinitely long bit
strings? Like infinite decimal
fractions but with bits
Claim:
::= {∞-bit strings}
is uncountable.
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Diagonal Arguments
Suppose
0
1
2
3
.
.
.
n
n+1
.
.
.
s0
s1
s2
s3
0
0
1
0
.
.
.
0
0
.
.
.
0
1
1
0
.
.
.
0
1
.
.
.
1
0
0
0
.
.
.
1
0
.
.
.
1
0
1
1
.
.
.
1
1
.
.
.
.
.
.
.
.
.
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.
1
.
0
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Diagonal Arguments
• Suppose
0
1
2
3
.
.
.
n
n+1
.
.
.
s0
s1
s2
s3
10
0
1
0
.
.
.
0
0
.
.
.
0
01
1
0
.
.
.
0
1
.
.
.
1
0
10
0
.
.
.
1
0
.
.
.
1
0
1
01
.
.
.
1
1
.
.
.
.
.
.
.
.
.
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01
.
01
.
10
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Diagonal Arguments
Suppose
1
⋯
0
…differs
from every row!
1
So
cannot be listed:
0
this diagonal
sequence
0
0
will be missing
1
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Cantor’s Theorem
For every set, A (finite
or infinite), there is no
bijection A↔P(A)
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There is no bijection A↔P(A)
Pf by contradiction: suppose
f:A↔P(A) is a bijection. Let
W::= {a A | a
 f(a)}, so for any a,
a W iff a  f(a).
f is a bijection, so W=f(a0), for some a0 A.
(∀a) a f(a0) iff a  f(a ).
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There is no bijection A↔P(A)
Pf by contradiction: suppose
f:A↔P(A) is a bijection. Let
W::= {a A | a
 f(a)}, so for any a,
a W iff a  f(a).
f is a bijection, so W=f(a0), for some a0 A.
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a f(a0) iff a  f(a ).
0
0
contradiction
0
13
So P(N) is uncountable
P(N)
= set of subsets of N
↔ {0,1}ω
↔ infinite “binary decimals” representing
reals in the range 0..1
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