Probability for a countable sample space

Cosider sample space, S, defined as

S = {1,2,3,4,5,6,7,8,9,................}

Experiment is to pick a number randomly from this set (with equal probability).

Question: How do you define probability measure for this?

Confusion I have -

Consider sets A1 = {1}, A2 = {2}, A3 = {3}, A4 = {4}, ..., An = {n},.... so on

Now let P(An) = x

P(S) = P(A1)+P(A2)+P(A3)+....+P(An)+..... (under the axiom of countable additivity)

LHS is 1 but RHS = 0 or infinite, so where is that I'm going wrong?