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Cantor's diagonal argument leads to a contradiction.

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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.

Parting words from the person who closed the correction: 
> The number of lines in the table is B^D (B to the D power). Not so. The number of lines used to construct the first D digits is simply D. The proof I give is quite standard. It may be helpful to consult some references.
Status: Rejected
Reference to the user who closed the correction.: 
Reference to the article this correction is about: 
Status of the article (was it accepted?): 
0
Status of the article (is it closed?): 
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What kind of correction is this: 
Error

Comments

You completely missed the point. I didn't claim that the number
of lines used to generate the diagonal was B^D. I'm claiming that
the number of lines in the largest set of real numbers between 0 and
1, including 0, not including 1, with D digits to the right of the
decimal is B^D. This is larger than the number of lines crossed by
the diagonal. So when a new number is generated from the diagonal,
it is a number that is in the set but not crossed by the
diagonal. This is still true when D goes to infinity. So Cantor's
diagonal argument doesn't work for this set and an infinite number
of other sets.
Generating a new number from the diagonal does work
for some sets. But the diagonal argument isn't claiming to be valid
for just those sets, it's claiming to be valid for all of them. Since
it doesn't work for all of them, it's not valid.
You don't have to refer me to other sources. I realize I'm tugging
on Superman's cape, that I'm saying the Kings naked, that I'm
breathing air way above my pay grade.

You are right. I have included your statement into https://www.hs-augsburg.de/ mueckenh/Transfinity/Transfinity/pdf Would you disclose your real name? Regards, WM

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