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.

The limit of your distributions is not a distribution so there's no problem.

If there's any sort of inconsistency in ZF or PA or any other major system currently in use, it will be much harder to find than this. At a meta level, if there were this basic a problem, don't you think it would have already been noticed?

Yes, but there's less reason for that. A big part of the problem with neutrinos is that since only a small fraction are absorbed, it becomes much harder to get good data on what is going on. For example, the typical neutrino pulse from a supernova is estimated to last 5 seconds to 30 seconds, while the Earth is under a tenth of a second in diameter. Gamma rays don't have quite as much of this problem and we can sort of estimate their directional data better.

On the other hand, the more recent work with neutrinos has been getting better and better at getting angle data which lets us get the same directional data to some extent.

You do know that both sets of ideas predate HPMOR, right?

That's a rule I'd strongly support other than in cases of absolutely unambiguous spamming or clear sockpuppets of banned individuals.

While I'm deeply concerned about the possibility that AA has been engaging in vote-gaming which does seem to be a bannable offense, it isn't clear to me that, as reprehensible as that comment is, that it is enough reason by itself for banning, especially because some of his comments (especially those on cryonics) have been clearly highly productive. I do agree that much of the content of that comment is pretty disgusting and unproductive, and at this point his focus on incel is borderline spamming with minimal connection to the point of LW. Maybe it would be more productive to just tell him that he can't talk about incel as a topic here?

Not only does it not predict such large swings it also doesn't fit with the fact that after such a swing (which occurs rapidly) he then gets a slow downward trend. I pointed this out to the moderators a while ago and so I have a record of how rapid some of the changes were:

http://lesswrong.com/lw/ls5/if_you_can_see_the_box_you_can_open_the_box/c1kf was at -9 within 8 hours of being posted, 12 hours later or so it was at +4. Note that it has now reverted to +0.

http://lesswrong.com/lw/ln8/february_2015_media_thread/bx5u was at -5, then within 24 hours went to +6 and is now +3.

http://lesswrong.com/lw/lli/open_thread_jan_26_feb_1_2015/bw6v was at -8 at 5 PM EST. At 7:10 EST it was at +6. In the same span http://lesswrong.com/lw/lli/open_thread_jan_26_feb_1_2015/bw6w was at -13 and went to +0. After the fact over the next few days, both those comments went into the deep negative. Similarly http://lesswrong.com/lw/lk7/optimal_eating_or_rather_a_step_in_the_right/bvmk was at -4, then went in the same 2 hour time span up to 3 and then went to 2 (so was left alone after that).

Curiously, within the same 2 hour time span as that set of rapid upvoting, two highly negative comments in support of A went through a similar swing with again a slow reversion over the next few days http://lesswrong.com/lw/lli/open_thread_jan_26_feb_1_2015/bw9t and http://lesswrong.com/lw/lli/open_thread_jan_26_feb_1_2015/bw7l

These aren't the only examples, but simply the most blatant

Based on this evidence I assign an extremely high credence that some form of karma abuse is going on with someone using multiple accounts (approximately 90% certain). I assign an 80% chance that this person is doing so deliberately to upvote comments which are seen at odds with "liberal" politics in some form. I assign a slightly over 50% chance that AA is doing this himself. The fact that it took until now for him to address such concerns despite the fact that others have mentioned them is not positive. After AA himself, I assign the next most likely individual to be Eugine for obvious reasons.