Re: infinity

>
> Noon may be un-reachable in a physical sense, but this is not a physical
> problem. It's a mathematical problem. The question makes perfect sense
> from a mathematical standpoint.
>
> Your final sentence above makes no sense at all. There is no such thing
> as a "last ball". All that is necessary to show that the vase is empty
> is to show that each ball is removed from the vase before noon and is not
> subsequently replaced.
>
> > Assume f(x) is an increasing function and f(0) > 0. Even though f(oo)
> > is undefined but I am quite sure that f(oo) cannot be 0.
>
> Why?
>
> Dave Seaman

Thanks Dave,

This is a paradox because it is not a physical nor a mathematical
problem but rather a logical problem!!!

Physically or mathematical it is not difficult to prove that the vase
is not empty at noon. Mathematically, # of balls in the vase can be
expressed by the equation f(t) = 9+9*log(1/t)/log(2) where t is the
time in minute before noon. 't' can be 1, 1/2, 1/4, 1/8, ... 1/OO.
f(1) = 9
f(1/2) = 18
f(1/4) = 27
f(1/8) = 36
Number of balls in the vase at noon is f(0) = OO. Try it :)

The statement: "All that is necessary to show that the vase is empty is
to show that each ball is removed from the vase before noon and is not
subsequently replaced." is a logical argument. Unfortunately, if
infinity gets involved, this statement alone is not sufficient to claim
the vase is empty before noon. The vase can only be claimed to be
empty at a particular time t is when all the balls are removed at that
time. At t = 1, ball 1 is removed but ball 10 (the last ball) is added
to the vase. At t = 1/2, ball 2 is removed but ball 20 (the new last
ball) is in the vase. There is no such time that all balls are
removed. Therefore, the vase cannot be empty.

If the problem is changed to remove the last ball each time then the
argument would be much simplier - ball 1 is always in the base.

.

Relevant Pages

  • Re: An uncountable countable set
    ... the vase, is consistent with the fact that no balls are removed at noon? ... The only relevant question is "According to the rules set up in the ... is each ball inserted before noon also removed before noon?" ... An affirmative answer confirms that the vase is empty at noon. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Does Han claim that there is any ball put in that is not taken out? ... the vase empties". ... But in order for the vase to transition from not-empty ... If the vase ever became empty, ...
    (sci.math)
  • Re: An uncountable countable set
    ... Does Han claim that there is any ball put in that is not taken out? ... the vase empties". ... If the vase ever became empty, ...
    (sci.math)
  • Re: infinity
    ... Number of balls in the vase at noon is f= OO. ... Unfortunately, if infinity gets involved, this statement alone is not sufficient to claim the vase is empty before noon. ... ball) is in the vase. ... You also keep ignoring my question. ...
    (sci.math)
  • Re: An uncountable countable set
    ... The only critical time dependency is that each ball to be inserted shall ... the vase is empty at noon of anything of any balls ... An affirmative answer confirms that the vase is empty at noon. ... given the times of insertions and removals. ...
    (sci.math)