Re: infinity


Tony Orlow (aeo6) wrote:
> David R Tribble said:
> > William Hughes wrote:
> > >> State E, noon. comes after all the steps. Since there are an infinite
> > >> number of things happening betweem any step at which there are balls
> > >> in the vase and state E, I don't have any problem with the vase
> > >> being empty at sate E. There is certainly no point at which the
> > >> vase suddenly becomes empty (it takes an infinite number of steps
> > >> to reach noon, so even times very close to noon are an infinite number
> > >> of steps away), and no last ball that is removed.
> > >>
> > >> Yes this is a bit wierd. But look at the definition of the set
> > >> of balls in the vase at state E, noon. This is very straighforward.
> > >> You either have to accept the counterintuitive result or
> > >> claim that the problem does not have an answer.
> > >>
> > Kirby Cook wrote:
> > > Or that something's missing. What? I don't know.
> > > But it seems to me that when you say that there are balls in the vase
> > > before noon, and no balls in the same vase at noon, but neither is there
> > > any *way* that the transition from balls to no balls is made, or can be
> > > defined, or even any point at which it happens, that something has been
> > > left out.
> >
> > It's the same thing as trying to say what happens when you add up
> > the infinite series:
> > s = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
> > s = 1
> >
> > At any point before you reach the end of the series, the partial
> > sum s(n) is less than 1. No matter how close you get, s(n) is not
> > quite 1. Only when you reach the "end" of the series does the
> > sum add up to 1. You can't define that magic "end" point when the
> > sum becomes equal to 1, but this is that "missing something that
> > happens" that you speak of.
> >
> > -drt
> >
> >
> Actually, the missing something that happens is the preservation of the ratio
> between balls inserted and balls removed. Intuition says that, out of an
> infinite number of balls, 9/10 have stayed in the vase, and 1/10 have been
> removed.

Definition says that the number of balls in the vase is the
number of balls in the set I\O.

The definition says nothing about the number of balls in I or the
number of balls in O (or for that matter about the number
of balls in I_n or O_n for any n). So it does not matter
what meaning we give to any of the phases "the number of balls in I"
"the number of balls in O", "the number of balls in I_n" or
"The number of balls in O_n" (We can use cardinality, TO's
definitons, arbitrary definitions, or declare that these phrases
have no meaning).

To answer the question, we just have to asign meaning to the
phrase "the number of elements in the set I\O". But the set I\O
is the empty set. The number of elements in the empty set is 0.

By definition the answer to the problem is 0.

Intutition (TO's, Kirby Cook's, mine, probably a lot of other
peoples as well) says that this answer is wrong. If a problem
has an answer that does not agree with intuition, the probem
is called counterintuitive.

>This ratio is disregarded by Cantor's method, which declares
> N = 9/10 N = 1/10 N, and therefore declares N-(1/10 N) = 0, instead of 9/10 N.
> This ratio should be preserved, and not disregarded, and then Kirby's
> intuitions will be satisfied

Kirby's, (yours, my, others) intuition cannot be satisfied
unless we change the definitions of I, O and the number of balls
in the vase at noon. Unfortunately, changing these definitions
is even more counterintuitive

>. and we can all be happy, and my son won't have to
> squinch up his face in confusion at a vase which inexplicably becomes
> instantaneously empty for no apparent reason!

I would not let your son near an attempt to do this experiment.
There is going to be a big explosion when you try to move ping
pong balls at arbitrarially high speeds.

-William Hughes

.

Relevant Pages

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  • Re: infinity
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  • Re: An uncountable countable set
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