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Misha comments on P(X = exact value) = 0: Is it really counterintuitive? - Less Wrong Discussion
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No finite amount of evidence will convince you. I can be convinced of infinitely unlikely things by an infinite amount of evidence just fine.
And if we're talking about a situation (like real life!) where you can't expect to receive an infinite amount of evidence, then we shouldn't be using probabilities of 0 or 1, either.