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For those of you who are
curious, we will go through an
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argument that establishes that
the set of real numbers is an
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uncountable set.
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It's a famous argument
known as Cantor's
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diagonalization argument.
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Actually, instead of looking
at the set of all real
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numbers, we will first look at
the set of all numbers, x,
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that belong to the open
unit interval--
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so numbers between 0 and 1--
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and such that their decimal
expansion involves
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only threes and fours.
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Now, the choice of three and
four is somewhat arbitrary.
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It doesn't matter.
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What really matters is
that we do not have
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long strings of nines.
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So suppose that this
set was countable.
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If the set was countable, then
that set could be written as
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equal to a set of this form,
x1, x2, x3 and so on, where
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each one of these is a real
number inside that set.
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Now, suppose that this
is the case.
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Let us take those numbers
and write them
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down in decimal notation.
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For example, one number
could be this one, and
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it continues forever.
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Since we're talking about real
numbers, their decimal
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expansion will go on forever.
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Suppose that the second number
is of this kind, and it has
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its own decimal expansion.
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Suppose that the third number
is, again, with some decimal
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expansion and so on.
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So we have assumed that our
set is countable and
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therefore, the set is equal
to that sequence.
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So this sequence exhausts all
the numbers in that set.
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Can it do that?
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Let's construct a new number
in the following fashion.
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The new number looks
at this digit and
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does something different.
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Looks at this digit, the second
digit of the second
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number, and does something
different.
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Looks at the third digit of
the third number and does
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something different.
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And we continue this way.
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This number that we have
constructed here is different
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from the first number.
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They differ in the
first digit.
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It's different from
the second number.
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They differ in the
second digit.
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It's different from the third
number because it's different
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in the third digit and so on.
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So this is a number, and this
number is different
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from xi for all i.
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So we have an element of this
set which does not belong to
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this sequence.
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Therefore, it cannot be true
that this set is equal to the
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set formed by that sequence.
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And so this is a contradiction
to the initial assumption that
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this set could be written
in this form, and this
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contradiction establishes that
since this is not possible,
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that the set that we have here
is an uncountable set.
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Now, this set is a subset of
the set of real numbers.
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Since this one is uncountable,
it is not hard to show that
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the set of real numbers, which
is a bigger set, will also be
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uncountable.
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And so this is this particular
famous argument.
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We will not need it or make any
arguments of this type in
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this class, but it's so
beautiful that it's worth for
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everyone to see it once
in their lifetime.