Re: infinity

On 5 Aug 2005 18:31:22 -0700, snapdragon31 wrote:

>> 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!!!

Saying that it's a "paradox" merely means that the result is
counterintuitive. It doesn't mean there is actually a contradiction.

> 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 :)

Incorrect. You are confusing f(0) with lim_{t->0-} f(t). The former is
0, but the latter is +oo. The former is what the problem asks for. Once
you get your notation straight, the paradox disappears.


> 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.

Incorrect. Change "at that time" to "at or before that time". Then the
time t=noon fills the bill.

> 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.

All balls are removed before noon.

> 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.

If the problem is changed so that all the balls are in the vase at 11:00
and from then on balls are only removed, never added, then do you agree
that it is possible for the vase to be empty at noon?


--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
.

Relevant Pages

  • Re: infinity
    ... Physically: Infinity is undefined physically. ... >> to show that each ball is removed from the vase before noon and is not ... >> the vase is empty before noon. ... The operation of adding or removal of balls is undefined at noon. ...
    (sci.math)
  • Re: An uncountable countable set
    ... To become empty means there is a change of state in the vase, from having balls to not having balls. ... There are always a specific number of balls, if additions and removals occur instantaneously. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Since the vase was empty to start with, it cannot later "become" empty after once having been empty, at least according to that definition. ... Noon does not exist in the experiment, or else you have infinitely numbered balls. ... insertion or removal or location of balls is a function of time. ...
    (sci.math)
  • Re: infinity
    ... >>> Physically or mathematical it is not difficult to prove that the vase ... # of balls in the vase can be ... this statement alone is not sufficient to claim ... >>> the vase is empty before noon. ...
    (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. ... then the sum becomes 0. ...
    (sci.math)