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Thomas comments on Infinitesimals: Another argument against actual infinite sets - Less Wrong
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Any individual real number has the zero probability, but at least one of them - is bound to happen.
One may or may not consider sub intervals. It is a side question. Just as rational numbers, or algebraic numbers on this interval. Every sub-interval has the probability equal of its length what is always nonzero. All rational numbers have the probability 0, for example.