To prove that the set of natural numbers does not have a one to one correspondence with that of the reals, we first need to assume the sets exist, and then move on to the diagnolisation argument. Why is this allowed but not shifting around numbers in divergent sums to get a value; every week there is at least one post here about some tricky way to find out the value of 1 + 2 + 3 +.. . Apparently addition is not well defined over divergent sums or something, but then how are infinite sets well defined. How do I figure out when playing around with infinity is fine and when it is not?