## General question on why Cantor used the diagonal method

Hi, I have a general question:
why specifically use the diagonal method to show that some sets have larger cardinality than that of the set of natural numbers?

I have to assume that the point was to show that one cannot order the sequence in any way which would correspond to a tie to the natural numbers, eg 1.1, 1.2 etc would still be placed in positions (eg) 1 and 2, but then the full set would take up all of the positions used by natural numbers and still allow room for many more positions for the fractions. But why should one use a diagonal to show there are many more possible positions, instead of any other method? Is this the simplest possible to think of, or does it have any specific use in other things? I mean intuitive it would be self-evident that even a fraction of a fraction of a fraction... of something would go on in a one to one tie to the natural numbers, so is there some use in coming up with a specific and easy to iterate set which can be fed back to the original (different in the diagonal) and still allow for the new set having space for more?

To me the diagonal presentation seemed somewhat similar to the first proof of there being more prime numbers than can be accounted for, so was wondering if it was just one possible way of making the argument or something inherently valuable due to ties to other parts of set theory.