Cantor's proof that the set of
points on a line segment is not countable.
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Restatement: If P1, P2, P3, ...
is a sequence of points on the line segment AB, then there is a point C
on the line segment that is not a member of this sequence.
[Thus all the points on the line cannot be counted by
any sequence.]
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Step 1. Cut the segment AB into
three segments of equal length. Choose either the left most or right most
of these three so that P1 is not in this third. Rename this segment A1B1.
So A1B1 is a segment contained
in AB and P1 is not in A1B1.
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Step 2. Cut the segment A1B1
into three segments of equal length. Choose either the left most or right
most of these three so that P2 is not in this third. Rename this
segment A2B2.
So A2B2 is a segment contained
in A1B1 and
P2 (and P1) is not in A2B2.
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Step k+1. Cut the segment AkBk
into three segments of equal length. Choose either the left most or right
most of these three so that P(k+1) is not in this third.
Rename this segment A(k+1)B(k+1).
So A(k+1)B(k+1) is a segment
contained in AkBk and Pk+1 is not in A(k+1)B(k+1).
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Now this process constructs a sequence of nested segments.
[In fact each segment is one third the length of the
previous segement. ]
So there is (exactly) one point that is common to all
these segments, which we will name, C.
And C is not any of the points in the original sequence
of points P1, P2, ....