Utilitarian2 comments on 0 And 1 Are Not Probabilities - Less Wrong
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Speaking of measure theory, what probability should we assign to a uniformly distributed random real number on the interval [0, 1] being rational? Something bigger than 0? Maybe in practice we would never hold a uniform distribution over [0, 1] but would assign greater probability to "special" numbers (like, say, 1/2). But regardless of our probability distribution, there will exist subsets of [0, 1] to which we must assign probability 0.
The only way I can see around this is to refuse to talk about infinite (or at least uncountable) sets. Are there others?