Re: infinity

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?
>
> 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, I am using basic math when speaking of numbers.
>
>

--
Smiles,

Tony
.

Relevant Pages

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