[FOM] An infinite paradox
liufs
liufs at x263.net
Thu Nov 21 06:17:53 EST 2002
In considering twin prime conjecture I encountered an infinite paradox:
‘demons, bucket and balls’ problem
There is a countable infinite number of consecutively numbered balls, a bucket,
and two demons.
Then for n=0,1,...
At time 1-(2^-(2n)) the first demon places the next 2 balls into the bucket.
At time 1-(2^-(2n+1)) the second demon removes the lowest numbered ball in the
bucket.
Q) At time t>1, how many balls are there in the bucket?
A1) The number of balls at time 1-(2^-(2n)) is ever increasing
=> an infinite number
A2) So the lowest numbered ball still in the bucket is?
=> no balls.
I think, from ever increasing we obtained the logical conclusion infinite,lim T'n=inf.
This conclusion is valid in the all probable worlds except the empty world.In the empty
world this conclusion would fail. Hence, we taken A2), no balls in the bucket,lim T'n=empty, and there is no contradiction.
My question is that if we known the bucket is not empty, lim T'n =/= empty,namely,we have removed the situation A2),from ever increasing, we obtained the logical conclusion infinite, lim T'n=inf, is it valid in the all probable worlds ? namely, have it finite counter-example? I appreciate an ardour man give me an answer or some references.
China,
Liu Fengsui.
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