> Well, I'm not at all sure that there's no problem with forward > supertasks. Surely, it is not difficult to come up with a > problematic case.
Yes, the worth of supertasks as indicators of philosophical concerns is very much up in the air. Some seem relevant, others just stupid. Perhaps (temporally) well-ordered supertasks are more sensible than most. But I doubt that's all there is to it.
One of my favourites is this; for naturals n:- compare...
a) At each time 1 - 1/n, add balls numbered 2^(n-1) to 2^n - 1 to the pot, and remove ball number n.
b) At each time 1 - 1/n, add 2^(n-1) - 1 balls to the pot, and replace the numbering stickers in agreement with case (a).
After time 1: the final situation in case (a) is that the pot is empty. in case (b), the pot has infinitely many balls with no stickers!
And yet at any intermediate time the two cases are indistinguishable!
This sort of example shows that even omega-supertasks can be remarkably silly!
> Now alter the situation slightly. At each step, again place 10 balls > into the vase and then remove one ball, but remove the ball > *randomly*. At the end, the vase may contain any number of balls
Actually NOT. The pot will be empty(!) [with probability 1]
For any ball, the probability of it being removed is like a harmonic series, which sums to oo, which by Borel-Cantelli means it will happen for sure. [meaning probability one, as always here]
HOWEVER - if you add (say) 1, 4, 9, 16... balls per turn, and again remove one at random, each turn, then (Borel-Cantelli) each ball has a positive probability of being left behind.
It is an interesting problem in probability generating functions to work out the individual proabilities for each ball, and the expected number, left in the pot at the end!