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?
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?