Problem with Cantors Diagonalization Proof.

[talanum1]

Cantor supposes we may make an infinite list of real numbers between 0 and 1 and pair them with Natural Numbers. Then he supposes we make another real number by changing digits along a diagonal in the list. We take the first digit of the first number and change it, take the second digit from the second number and add one to it, and so on. Then he says that we constructed a number not on the list.

My objection is that in order to conclude this we must be able to state that the digit at infinity of the infinite numbered real number got taken and changed. But "digit at infinity" and "infinite numbered real number" is undefined.

One can also not say: "change the n'th digit by adding one and then let n tend to infinity." because the "tending to" operations operand must be specified and the process will never end: the operand at infinity must actually be evaluated.

We don't have access to the digit at infinity.

This is especially acute because we have wrong intuition about infinity.

[talanum1]

The process of taking the n'th digit and adding 1 and then letting n tend to infinity would look like;

one time we get the changed digit is 1 and another time we get it is 2, so we have a contradiction so it must be impossible to change the n'th digit and then letting n tend to infinity.

[Dave A]

For those of us seeking assurance that this means precious little in practical application to our lives, you can get an insight into the uber-hypothetical numerical nature of this construct here.

Dear Talanum1, please give a practical example of how this is of value in real life in terms us simpletons can understand.

[talanum1]

For counting the amount of dots in a line.