It is, and if you choose the number uniformly at random, it's not just "effectively" zero, it is precisely zero.
GP's point, as I understand it, is that it is not actually possible to choose a number from [0, 1] uniformly at random in "real life".
I think you could argue that, e.g. in the dartboard thought experiment, the probability of choosing individual points doesn't really matter: only probabilities of measurable subsets with positive measure matter.
I guess, but, the set is not even countably infinite. "Selecting at random" is something that happens in the real, non-infinite world, not in the mathematically rigorous would where infinities can exist. So, no, probability-zero events do not happen in either.
Not necessarily - you might just come up with a number you know is in the set, say pi/4. I know it's in the set because it satisfies the conditions that define it. Still, the odds of that particular number being picked up are zero.
It wouldn't "select" as if it were an infinite deck of cards, but rather generate a number we know is on the infinite set. It can very well take an infinite amount of time to come up with the digits though...
GP's point, as I understand it, is that it is not actually possible to choose a number from [0, 1] uniformly at random in "real life".
I think you could argue that, e.g. in the dartboard thought experiment, the probability of choosing individual points doesn't really matter: only probabilities of measurable subsets with positive measure matter.