I'm completely flummoxed by the level of discussion here in the comments to Unknowns's post. When I wrote a post on logic and most commenters confused truth and provability, that was understandable because not everyone can be expected to know mathematical logic. But here we see people who don't know how to sum or reorder infinite series, don't know what a uniform distribution is, and talk about "the 1/infinity kind of zero". This is a rude wakeup call. If we want to discuss issues like Bayesianism, quantum mechanics or decision theory, we need to take every chance to fill the gaps in our understanding of math.

To answer your question: [0,1] does have the same cardinality as all reals, so in the set-theoretic sense they're equivalent. But they are more than just sets: they come equipped with an additional structure, "measure". A probability distribution can only be defined as uniform with regard to some measure. The canonical measure of the whole [0,1] is 1, so you can set up a uniform distribution that says the probability of each measurable subset is the measure of that subset (alternatively, the integral of the constant function f(x)=1 over that subset). But the canonical measure of the whole real line is infinite, so you cannot build a uniform distribution with respect to that.

After you're comfortable with the above idea, we can add another wrinkle: even though we cannot set up a uniform distribution over all reals, we can in some situations use a uniform prior over all reals. Such things are called improper priors and rely on the lucky fact that the arithmetic of Bayesian updating doesn't require the integral of the prior to be 1, so in well-behaved situations even "improper" prior distributions can always give rise to "proper" posterior distributions that integrate to 1.

Yes, it's uncountable, so the probability of any particular real number is zero. I'm not going to go into detail by what I mean by a probability distribution here (though I'll note that cousin it used the word "integrate"). To uniformly pick a number $x$ from 0 to 1, use an infinite sequence of coin flips to represent it in binary. Infinity is impractical, but for natural questions, like: is $x$ between 1/5 and 1/sqrt(2), you will almost surely need only finitely many flips (and your expected number is also finite). And the probability that the answer is yes is 1/sqrt(2)-1/5; that is the sense in which it is uniform.

You can transfer this distribution to the set of all real numbers, eg, by logit or arctan, but it won't be uniform in the same sense. One can satisfy that uniformity by a measure which not a probability measure.