nice expose, Ken!
how about you leave 1/2 ball in the bag: at the kth time step, change the count of balls in the bag by (-1)^k, i.e., put a ball in the first step, remove one ball in the second step, add one at the third step. 1-1+1-1+... = 1/2, or:
leave -1/12 balls in the bag: at the kth time step, add k balls to the bag. 1+2+3+... = -1/12.
what breaks the balls into pieces to make this happen? what if the balls are 'atoms' ('indivisible')?
Does that mean that the process is illogical for indivisible things? Or, does it mean that nothing is indivisible? If 'balls' were instead distinct fixed-volume elements of space, and the bagging operation was merely the process of labeling such elements, would the process generate an 'address book' for the entire universe? Does space need to be infinitely divisible for this to be logical?

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