I thought the writeup was convincing and addresses the heart of the paradox. The only nitpick: you can have uniform probability distributions on infinite sets, like [0,1]: https://en.wikipedia.org/wiki/Continuous_uniform_distributio... There p(x) = 0 for any x, but for fixed e, p(x +/- e) is the same for all x.
But you can't have such a distribution on an unbounded set, which is where the paradox fails. If we had a uniform distribution on an unbounded set, p(x +/- e) has to be the same for all x and therefore nonzero, but
p(1 +/- e) + p(2 +/-e) + ...
has to sum to <= 1. It is an infinite sum of nonzero terms so this is a contradiction. (The same argument works if you drop the epsilon for thinking of a distribution on the integers).
I think your writeup was basically clear on this in terms of the math, just some of the language was a bit confused.
Yeh, U[0, 1] is different because it assigns non-zero probabilities to intervals, not points. In this case we're assuming that we live in an uncountable population (each real in [0, 1] is a person), so you can't do things like assign a unique number to each person. There, even if the maniac goes on kidnapping forever, he will only kidnap a countable subset of the population. Thinking about this honestly makes my brain hurt a little.
But you can't have such a distribution on an unbounded set, which is where the paradox fails. If we had a uniform distribution on an unbounded set, p(x +/- e) has to be the same for all x and therefore nonzero, but
has to sum to <= 1. It is an infinite sum of nonzero terms so this is a contradiction. (The same argument works if you drop the epsilon for thinking of a distribution on the integers).I think your writeup was basically clear on this in terms of the math, just some of the language was a bit confused.