Yakk wrote:You have a supply of numbered balls.
You start with ball 0 in the jug.
At T_i, you place balls (10i+1, ..., 10i+9) into the jug. You then, in the jug, renumber the lowest numbered ball to 10i+10.
At midnight, there are no balls in the jug. No balls have been removed from the jug, and an infinite number of balls have been added to the jug.
Why on earth would that be? At midnight in your situation, there ought to be infinitely many balls with undefined labels. If you change it slightly so that you start with ball 1, then at midnight there will be 1 ball for each natural number not terminating in 0, but each will bear that natural number followed by infinitely many 0s. (Or, be undefined, depending on how you are doing the labeling, but if you just append 0s as the modification, this is what you get). This solution fits perfectly with the theory that gives an empty jug in the original case - I look at each ball and sees what happens to it.
I was presuming reasonable (yet unspecified) axioms. To be explicit, if not formal:Axiom of Inclusion
A ball with label L is said to be in the jar if, for all n from N, for all a > n the ball is in the jar after T_a.Axiom of Exclusion
A ball with label L is said not to be in the jar if there exists an n from N such that for all a > n, no ball with the label L is in the jar after T_a.Natural Axiom
All balls in the jar have natural number axioms.Axiom of Excluded Middle
A ball is in the jar, or not in the jar, at each time, and at midnight.Axiom of Count by Cardinality
The number of balls in the jar is the cardinality of the set of balls in the jar.
(And yes, more axioms are needed to finish the job).
These are all reasonable axioms -- and they lead directly to "yes, if you remove the lowest numbered ball while adding 10 higher numbered ones, you end up with zero balls in the jar at midnight". And similarly removing the highest numbered jar results in an infinite number of balls in the jar at midnight.
They are not, however, the only
axioms you can choose for this problem. And your choice of axioms is both important, and not pre determined
for this kind of problem. We pick reasonable axioms, and if there is a problem we examine why the problem is generated -- and see if we can find better axioms that are consistent, interesting, and don't generate the problem!