Does cantor's diagonal argument to prove uncountability of a set and its powerset work with any arbitrary long column or row rather than the diagonal?

Question

Does cantor's diagonal argument to prove uncountability of a set and its powerset work with any arbitrary column or row rather than the diagonal?

Does the diagonal have to be infinitely long or may it consist of only a fraction of the length of the infinite major diagonal?

Even if we picked a finitely long diagonal, wouldn't that still be used to discover a new number via a function that is not counted by the naturals?

Answer 1

To answer your question, think about what makes Cantor's proof work. You need an infinite set of positions (columns) where you can modify the entry in that row to make it differ from the one you're constructing. You have to eliminate every row.

So the argument works for any infinite sequence of columns $c_1 < c_2 < c_s < \cdots$.

The simplest such sequence is the diagonal $1,2,3, \cdots$ which is why it's always used.