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Sarunas comments on Open Thread, Jul. 27 - Aug 02, 2015 - Less Wrong Discussion
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There is no such thing as a uniform probability distribution over a countably infinite event space (see Toggle's comment). The distribution you're assuming in your example doesn't exist.
Maybe a better example for your purposes would be picking a random real number between 0 and 1 (this does correspond to a possible distribution, assuming the axiom of choice is true). The probability of the number being rational is 0, the probability of it being greater than 2 is also 0, yet the latter seems "more impossible" than the former.
Of course, this assumes that "probability 0" entails "impossible". I don't think it does. The probability of picking a rational number may be 0, but it doesn't seem impossible. And then there's the issue of whether the experiment itself is possible. You certainly couldn't construct an algorithm to perform it.
Given uncountable sample space, P(A)=0 does not necessarily imply that A is impossible. A is impossible iff the intersection of A and sample space is empty.
Intuitively speaking, one could say that P(A)=0 means that A resembles "a miracle" in a sense that if we perform n independent experiments, we still cannot increase the probability that A will happen at least once even if we increase n. Whereas if P(B)>0, then by increasing number of independent experiments n we can make probability of B happening at least once approach 1.