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Cantor's diagonal argument leads to a contradiction.
For the proof to be valid the diagonal is required to cross all lines in the two-way infinite table. If C is defined as the ratio of the number of lines crossed by the diagonal divided by the number of lines in the table, C is be required to be equal to 1.
The base of the number system used in the proof is usually 10, so define B=10. (The arguments work for other bases too.) The number of lines crossed by the diagonal is the same as the number of digits in the new element being generated. Define this as D. The number of lines in the table is B^D (B to the D power). The value of C can be defined as C=(D/(B^D)).
As D goes to infinity the value of C goes to zero. This contradicts the requirement that C=1. The proof is invalid.
To understand this look at finite values of D.
If D=1 then the diagonal crosses 1 out of the 10 lines, so that C=0.1.
If D=2 then the diagonal crosses 2 out of the 100 lines, so that C=0.02.
If D=3 then the diagonal crosses 3 out of the 1000 lines, so that C=0.003.
As the width of the table increases the length of the diagonal increases at the same rate. But the height of the table increases much more rapidly. It is not possible for the diagonal to catch up to this increase even when D goes to infinity. C does not equal 1 for any values of D.
This doesn't mean that reals are not uncountable. It means that this particular arument is not a proof of it.
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