I think the crux of Divine Insight's error is that he/she continues to insist that the number of nonzero digits must dictate the representation be rectangular. This is an invalid assumption.

There is no valid reason one cannot simply add zeroes to DI's 'rectangular' representations and still use the same set of actual numbers.

Another problem invaliding DI's argument is that DI refuses to acknowledge something as basic as 0.333 . . . =1/3 or that root 2 is a valid real number, then none of his arguments have anything to do with the actual argument Cantor is making, not to mention vast other areas of mathematics.

He gives no valid reason for this rejection. He simply declares such notions as absurd or uses similar subjective or vague language to dismiss this.

And the further problem is that he assumes things at the finite level must be equally valid at the infinite level. Again, this is an invalid assumption. I believe, for example, he refused to address that an infinite set can have propers subsets of the same cardinality while this cannot be true of finite sets.

No you have not. You have simply insisted repeatedly that your representation is the only valid one by fiat.Divine Insight wrote:1. I have shown why any compete list of numerals must necessarily contain more rows than columns, and can therefore not be treated as a square list.

You continue to have an ambiguous definition of 'complete' which it seems applies only to the representation, the numerals, and not the actual numbers. When I changed the representation, you insisted I had to change the set. Again, no valid basis for this claim.

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is a complete list of the set (0, 1, 2, 3} expressed in binary notation. Complete, as understood in Cantor's proof, means the entirety of a set assumed to be countably infinite. If DI wants to say complete should apply to other types of sets, that is possible, but the term should mean 'the entirety of a given set' in some sense, or it has nothing at all to do with Cantor's proof, which would mean, as I showed above, DI's argument is a straw man.

If DI wants to insist instead that 'complete' means all possible numbers of a given number of digits using a particular representation, that could be used, but then his argument is irrelevant to Cantor's argument.

If we used base two representations instead of base 10 representations in Cantor's proof, the number of nonzero digits in some of the numbers in the list (whichever ones had a finite number of nonzero digits) would change, but the argument would be the same, just using only 0 and 1 as digits.

Your rows versus column claim only is valid if you stick to your invalid assumption that your rectangular representation scheme is the only one. If you stick to this assumption, then you are dealing with representations as the key concept, not actual sets of numbers. Any of your finite representations could be made to be square by adding a finite number of zeroes. You refuse to acknowledge this, without basis in logic or fact. It does not matter if the different representation could represent more numbers at the finite level. The different representation still represents the same set original set of numbers. The new number created by going down the diagonal will still not be in the original set.2. I have also demonstrated that it is the nature of these numerical lists to grow exponentially in the number of rows required to record every numeral with every column appended.

Do you disagree with this? If so place address that point.

But really, the big problem is DI's refusal to accept the logical validity of infinite decimal representations.

DI labels a claim as absurd without any actual valid refutation, and the claim itself continues the 'squareness' straw man argument..3. I have shown that crossing off digits using a diagonal line that crosses off the next digit in the next row necessarily creates a square list of crossed off numerals. Therefore that list cannot be a complete list of numerals, nor can it be used to prove what may or may not be on a complete list of numerals.

Do you disagree with this? Is so please explain why without resorting to the absurd claim that when taken to infinity a complete list of numerals will magically become square. Where is there any rationale for such an absurd claim? I have already shown that completed lists must necessarily grow exponentially in rows with every digit column added. Therefore there is no justification for claiming that these innately rectangular lists will magically become square at infinity. The situation only gets worse with every digit crossed off the list. What's going to magically make the situation get better as you approach infinity?

The claim that a 'square list of crossed off numbers' is created when you cross off a single number in each row is at best unclear if not nonsensical.

And again, 'square' is a property of the representation, not the set of numbers itself. It is an invalid concept with respect to Cantor's proof which does depend on possibly infinite decimal representations but does not depend on this vague notion of 'squareness' which would not even seem to make sense given DI is applying it to an 'infinite by infinite' square.

DI creates the ambiguous notion of squareness at infinity and then berates mathematicians for 'not understanding' his ad hoc notion. Neither Cantor nor any other mathematicians I am aware of makes claims about 'squareness at infinity' in this context. This is DI's straw man rearing its head again.You need to explain how that works. You can't just claim that it will magically become square at infinity. Mathematics doesn't stand on unsubstantiated claims, or does it?

DI has not demonstrated this, but seems unwilling or incapable of accepting that.

Therefore I have successfully demonstrated why Cantor's diagonal proof does not prove what he claims.

Since DI's argument hinges on insisting without a valid reason that a representation like

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must be considered invalid in favor of his rectangular representation, even though both represent the same set, his argument is an invalid straw man.

Consider, for example, that while Cantor's proof typically is phrased in terms of the real numbers between 0 and 1, it could just as easily been phrased in terms of the real numbers between N=10^(1000^1000) and N+1. We just add 1 and the appropriate number of zeroes ahead of the decimal point. This would make, in the same vague sense that DI insists Cantor's list be square, the representation rectangular but wider than it is long by the number of digits in N.