Is it possible to choose uniformly from $(0,1)$? 
Can a method be constructed to choose one number uniformly from $(0,1)$?

It has been argued that an event which is possible can have zero probability - such as the probability of selecting any given number when selecting uniformly from $(0,1)$.
But I hold that an event with zero probability can never happen, and therefore I conclude that it is impossible to choose uniformly from  $(0,1)$.
Can a method be constructed to choose uniformly from $(0,1)$?  I've a feeling that if there is a method, there's no guarantee it ever stops.
 A: If by "method" you mean an algorithm (that can be implemented on a computer), then obviously the answer is "no" for the trivial reason that computers cannot deal with real numbers, since there are uncountably many reals but only countably many finite strings, which are all that a computer can deal with.
Furthermore, it does not help to restrict to computable reals, because it is just impossible to define a uniform distribution on computable reals. Basically, each must have zero probability because there are infinitely many of them, but then countable additivity implies that their total probability is zero.

But I hold that an event with zero probability can never happen, and therefore I conclude that it is impossible to choose uniformly from $(0,1)$.

So, yes, in the real world you can never hope of choosing a uniform random real from $(0,1)$, even if you can somehow obtain unbiased random coin flips.

That said, the mathematical notion of probability and possibility is not the same as any real-world notion. For example, mathematically it is possible to have an infinite sequence of fair coin flips, and hence it is possible for the entire sequence to consist of heads even though that sequence has zero probability. It does not conflict with actually tossing a fair coin in real life (even if there is a fair coin), because we can never toss it infinitely many times.
Mathematical probability is of course based on mathematical definitions, so it is largely irrelevant whether statements in probability theory have a meaningful interpretation in the real world. It is actually not so bad for countable discrete processes such as the coin flip. A probability of zero for a class of sequences means that in the real world you can be sure of eventually escaping that class. For instance, you can be sure that eventually the coin will come up tails, even though you cannot predict when that happens and may have to wait an indeterminable amount of time.
