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Re,Two infinite sets

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  • liufs
    I had readied the discussion above. We encountered¡®demons, buckets and balls¡¯ problem of Phil again.#3932. ... Now Phil s wisdom chooses Lim Tn =
    Message 1 of 1 , Oct 29, 2002
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      I had readied the discussion above.
      We encountered‘demons, buckets and balls’ problem of Phil again.#3932.

      >I think in summary the problem lies in the same category as the two demons with
      >their bucket of balls:

      >There is a countably 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 Phil's wisdom chooses Lim Tn = empty.This is right.

      I think,it is more interesting that
      this is one and only one admittable 'counterexample'of infinity.


      Obviously,The number of balls is increasing in bucket
      Lim | Tn | = inf,
      then demon removes the lowest numbered ball in the bucket iff the bucket is empty
      Lim Tn = empty.

      Essentially, this demon's problem is a contradiction of the theory about the predicate:
      R(k,a), i.e. the ball in the bucket,
      Lim Tn = { a: R(k,a)}.
      A1, | Lim Tn | = inf,
      A2, | Lim Tn | = 0.
      By Godel's completeness theorem,
      A theory is contradictable iff it have not any model,(it is a second order theorem).
      so that,the predicate R(k,a) have no model, namely predicate R(k,a) have no any interpretation

      { a: R(k,a)} = empty, Lim Tn = { a: R(k,a)} =empty.


      But, If we know lim Tn =/= empty, we know at least there is a ball in bucket, then the second
      demon does not remove the lowest numbered ball in the bucket, then the predicate
      R(k,a) have a model
      { a: R(k,a)} =/= empty.
      then the contradiction of the theory about predicate R(k,a) would not admittable. We have proved
      Lim | Tn | = | lim Tn | = inf.

      For the k-tuple primes, the pattern has given a model of predicate R(k,a), so that they are
      infinite. There is not any counterexample.
      Perhaps, this is the mystical veil of the twin prime.


      China
      Liu Fengsui.


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