Re: infinity

In article <MPG.1d654dac9602b11398a05a@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Jesse F. Hughes said:
> > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> >
> > > Jesse F. Hughes said:
> > >> Let's change the problem slightly. Again, we have an infinite set of
> > >> ping pong balls, each ball labeled with a natural number. But instead
> > >> of the old procedure, let's put *all* of the balls into the vase at
> > >> 11:59 and remove the first one. At 11:59:45, we remove the second,
> > >> and so on.
> >
> > > This is the same question Virgil posted a half dozen times in a row.
> >
> > Yes it is, but that is coincidental.
> >
> > >>
> > >> Tony: Is the vase empty or not at noon?
> > > Yes.
> > >>
> > >> If empty, then when was the last ball removed?
> > > Noon.
> > >>
> > >> If not empty, then which balls did we fail to remove?
> > > None.
> > >>
> > >> Can we put infinitely many balls into a vase by doing it one at a time
> > >> (with increasing speed)? If so, are we able to also empty a vase with
> > >> infinitely many balls by the same method? (Countably infinite in each
> > >> case, of course.)
> >
> > > Yes, but not all countable infinities are the same. This is a basic
> > > problem
> > > with "cardinality".
> >
> > But in both examples (the ten-at-a-time example and the all-at-once
> > example), we put the same number of balls in the vase: one ball for
> > each natural number. And in both examples, we removed the balls in
> > exactly the same way. How can the outcome be any different?
> You didn't remove the balls the same way. In the original example, every time
> you removed a ball you added ten. How can that be ignored? If you started
> with
> an infinite number, then removed one and added ten, you would NOT have zero
> at
> noon. Or, would you?

Depends on the precise rules for doing it.
> >
> > Or do you think that putting in balls ten at a time until we've
> > exhausted the set N of natural numbers produces a *bigger* set than
> > putting all of the balls in the vase at once? Where did those extra
> > natural numbers come from?
> No, I think that adding ten and removing one is the same as adding nine, but
> silly me

Now those last tow words we can all agree on.
.

Relevant Pages

  • Re: infinity
    ... >>> the vase keeps growing as you approach noon. ... the algorithm which describes the filling of the vase with balls ... Start with an empty vase. ... we try to connect some logical reasoning (about putting ...
    (sci.math)
  • Re: infinity
    ... the set of balls in the vase at state E ... >> consists of a finite number of sets or an infinite number of sets. ... The sum of an infinite series does depend on the number ...
    (sci.math)
  • Re: infinity
    ... by definition the vase is empty at state E. ... the set of balls in the vase at state E ... When you say "There is no change whether or not there are an infinite number of sets I_n", ... given by the sum of an infinite series. ...
    (sci.math)
  • Re: An uncountable countable set
    ... -1/n, where n is a natural number, there are balls in the vase. ... Let S be the set of naturals on balls removed before noon. ...
    (sci.math)
  • Re: infinity
    ... >> I is the union of a bunch of sets. ... >> Define I_n to be the set of balls added at step n. ... by definition the vase is empty at state E. ... > are an infinite number of sets I_n", ...
    (sci.math)