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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.
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.