Your doubts about Cantor's diagonal argument touch on deep philosophical and mathematical points about infinity and the nature of infinite lists. Let's unpack this carefully.
Key points in your objection:
You focus on the fact that an "infinite numbered" real number and the "digit at infinity" are not well-defined concepts in standard mathematics.
You note that the diagonal construction depends on changing the nth digit of the nth number, for all n, and question what it means to do this "at infinity."
You observe that there is never a "final" or "bottom" of the infinite list that one could reach to finish the diagonalization, so how can the new number constructed be claimed to differ from every number?
How Cantor's argument actually addresses this:
Infinite lists and infinity as a completed totality:
Cantor works with infinite sequences as completed objects, not as a process or until reaching a last element. The list is conceptually infinite but assumed as a complete entity indexed by natural numbers.
Constructing the new number digit-by-digit:
The construction of the new number does not require reaching a "digit at infinity". Instead, for each finite position nnn, we specify the nnnth digit of the new number by looking at the nnnth digit of the nnnth number on the list and changing it. This defines the new number fully as an infinite sequence of digits — each digit is defined at a finite step.
No need to "complete" the process physically:
The proof is not about actually traversing all digits; it relies on the nature of infinite sequences — which are accepted as mathematical objects distinct from finite sequences. The infinite object is given by its specification for every finite position.
Why the new number differs from every listed number:
Since the new number differs at digit nnn from the nnnth number on the list, it cannot be identical to any number in the list — because to be identical, all digits would have to match.
On the "list growth" or "shape" analogy:
The diagonalization proof does not require actually "reaching the bottom" of the list. Instead, it shows that if such a complete infinite list of all numbers existed, we could construct a number not on it. This is a logical contradiction, so no such complete list can exist.
Summary
Your concern relates to intuitions about infinity as a process or physical traversal — but Cantor's argument treats infinite sequences as completed objects, each digit defined at a finite index. The concept of infinite sequences and their use in modern set theory bypasses the idea of "digit at infinity" — it works with each digit individually, all digits specified by a rule.
sources:
1.
Cantor's Diagonalization Proof of the uncountability of the real numbers
2.
What is Wrong with Cantor’s Diagonal Argument?
3.
Cantor and Contradictions | Dave Kilian's Blog
4.
5.
Controversy over Cantor's theory - Wikipedia
6.
Cantor’s Diagonal Proof and various misconceptions as to what it actually proves
