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demons, buckets and balls

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  • liufs
    ... = an infinite number ... = no balls. ... Now I discovered that the trap is Q) At time t 1, how many balls are there in the bucket?. We modify the Q) to
    Message 1 of 1 , Dec 1, 2002
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      I try look for the trap, which is hiding in Phil's "demons, buckets and balls":

      >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.
      >This calls for wisdom: let him who has understanding reckon the number of the balls...

      Now I discovered that the trap is
      Q) At time t>1, how many balls are there in the bucket?.
      We modify the Q) to be
      Q') At time t>1, whether the bucket is empty or not?
      Without so much wisdoms , we can check mechanically the formal proof:
      Then for n=0,1,...
      At time 1-(2^-(2n)) the first demon places the next 2 balls into the bucket.

      Namely, we suppose that
      the bucket is not empty.

      At time 1-(2^-(2n+1)) the second demon removes the lowest numbered ball 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.

      A contradiction, so that we proved
      the bucket is empty.

      If we had known that the bucket is not empty, namely we had known the pattern of the k-tuple prime, then we ask
      Q') At time t>1,how many balls are there in the bucket?
      By ever increasing, we proved the number of k-tuple primes is infinite.
      There is no any counterexample.

      Now I wish,understanding my proof of twin prime conjecture
      http://www.primepuzzles.net/conjectures/twinconjliusproof.doc
      will be more easy.

      China.
      Liu Fengsui.
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