The logic of the doomsday paradox would predict, from any example of a positive integer however large, that with probability 1 there can be only finitely many positive integers. Since there are infinitely many positive integers the logic must be wrong.
The doomsday paradox is the sort of reasoning about population dynamics one might expect out of the philosophy department. In the biology department the reasoning is more along the lines of http://en.wikipedia.org/wiki/Population_dynamics, with the Lotka-Volterra equation playing an important role.