## Re,Two infinite sets

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• 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
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.