Comment on Open thread, Oct. 12 - Oct. 18, 2015

Mon, 12 Oct 2015 20:34:24 GMT

Phrasing it as a "super-task" relies on intuitions that are not easily formalized in either PA or ZFC. Think instead in terms of a limit, where your nth distribution and let n go to infinity. This avoids the intuitive issues. Then just ask what mean by the limit. You are taking what amounts to a pointwise limit. At this point, what matters then is that it does not follow that a pointwise limit of probability distributions is itself a probability distribution.

If you prefer a different example that doesn't obfuscate as much what is going on we can do it just as well with the reals. Consider the situation where the nth distribution is uniform on the interval from n to n+1. And look at the limit of that (or if you insist move back to having it speed up over time to make it a supertask). Visually what is happening each step is a little 1 by 1 square moving one to the right. Now note that the limit of these distributions is zero everywhere, and not in the nice sense of zero at any specific point but integrates to a finite quantity, but genuinely zero.

This is essentially the same situation, so nothing in your situation has to do with specific aspects of countable sets.

Discuss