Suppose someone described a lottery with three tickets, where it was certain that some ticket won, and the probability of each ticket winning was 1/100. We could note that the description violates additivity. Or, more intuitively, we could say that while lottery would make sense with, say, win probabilities 1/3, 1/3 and 1/3, the claimed 1/100 is just way too small a probability, because it is so much smaller than a set of probabilities that do make sense for a lottery with these many tickets.
We can say the same thing about infinite countable lotteries with infinitesimal outcome probabilities. Suppose the proposed probability of each individual ticket winning is some infinitesimal u. Now consider a perfectly probabilistically sensible and unparadoxical, albeit unfair, infinite lottery with individual win probabilities 1/2, 1/4, 1/8, 1/16, .... That lottery makes perfect sense. But our alleged infinitesimal probability lottery has the same number of tickets, but assigns to each one an infinitely smaller probability, since u is infinitely smaller than 1/2n. And so our alleged infinitesimal probability lottery assigns much too small a probability to each ticket.