I’m not sure what an “infinite set atheist” is, but it seems from your post that you use different notions of probability than what I think of as standard modern measure theory, which surprises me. Utilitarian’s example of a uniform r.v. on [0, 1] is perfect: it must take some value in [0, 1], but for all x it takes value x with probability 0. Clearly you can’t say that for all x it’s impossible for the r.v. to take value x, because it must in fact take one of those values. But the probabilities are still 0. Pragmatically the way this comes out is that “probability 0″ doesn’t imply impossible. If you perform an experiment countably-infinitely many times with the probability of a certain outcome being 0 each time, the probability of ever getting that outcome is 0; in this sense you can say the outcome is almost impossible. However it’s possible that each outcome individually is almost impossible, even though of course the experiment will have an outcome.
You can object that such experiments are physically impossible e.g. because you can only actually measure/observe countably many outcomes. That’s fine; that just means you can get by with only discrete measures. But such assumptions about the real world are not known a priori; I like usual measure theory better, and it seems to do quite a good job of encompassing what I would want to mean by “probability”, certainly including the discrete probability spaces in which “probability 0″ can safely be interpreted to mean “impossible”.
You’re right, it’s not that hard to come up with larger countable classes of reals than the computables; I just meant that all of the usual, “rolls-off-the-tip-of-your-tongue” classes seem to be subsets of the computables. But maybe Nick is right, and the definables are broader. I haven’t studied this either.
And yes, I also sometimes think about how assumptions I make about life and the perceptible universe could be wrong, but I do not do this much for mathematics that I’ve studied deeply enough, because I’m almost as convinced of its “truth” as I am of my own ability to reason, and I don’t see the use in reasoning about what to do if I can’t reason. This is doubly true if the statements I’m contemplating are nonsense unless the math works.
Eliezer:
I’m not sure what an “infinite set atheist” is, but it seems from your post that you use different notions of probability than what I think of as standard modern measure theory, which surprises me. Utilitarian’s example of a uniform r.v. on [0, 1] is perfect: it must take some value in [0, 1], but for all x it takes value x with probability 0. Clearly you can’t say that for all x it’s impossible for the r.v. to take value x, because it must in fact take one of those values. But the probabilities are still 0. Pragmatically the way this comes out is that “probability 0″ doesn’t imply impossible. If you perform an experiment countably-infinitely many times with the probability of a certain outcome being 0 each time, the probability of ever getting that outcome is 0; in this sense you can say the outcome is almost impossible. However it’s possible that each outcome individually is almost impossible, even though of course the experiment will have an outcome.
You can object that such experiments are physically impossible e.g. because you can only actually measure/observe countably many outcomes. That’s fine; that just means you can get by with only discrete measures. But such assumptions about the real world are not known a priori; I like usual measure theory better, and it seems to do quite a good job of encompassing what I would want to mean by “probability”, certainly including the discrete probability spaces in which “probability 0″ can safely be interpreted to mean “impossible”.
You’re right, it’s not that hard to come up with larger countable classes of reals than the computables; I just meant that all of the usual, “rolls-off-the-tip-of-your-tongue” classes seem to be subsets of the computables. But maybe Nick is right, and the definables are broader. I haven’t studied this either.
And yes, I also sometimes think about how assumptions I make about life and the perceptible universe could be wrong, but I do not do this much for mathematics that I’ve studied deeply enough, because I’m almost as convinced of its “truth” as I am of my own ability to reason, and I don’t see the use in reasoning about what to do if I can’t reason. This is doubly true if the statements I’m contemplating are nonsense unless the math works.