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.