Re: infinity

Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:

> 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?

But we removed the same number of balls. In fact, in both cases, we
removed the same ball at each time. So how come there are balls in
the vase at the end of the original (ten-at-a-time) case but it's
empty in the second (all-at-once) case? (Below, you admit that the
same set of balls are put in the vase.)

Surely it can only be that more balls were put in the vase in the
first case than in the second. Else, there would be balls left over
in the first case only if there were balls left over in the second.

Let A be the set

{ n in N | ball n was put in the vase at some time prior to noon
in the ten-at-a-time case }

Let B be the set

{ n in N | ball n was put in the vase at some time prior to noon
in the all-at-once case }

We can write:

A = { n in N | there is some k in N such that 10 * k > n }

B = N.

Now, in both cases, we removed the same ball at the same time, so let

C = { n in N | ball n was removed from the vase prior to noon }
= N.

Clearly, the number of balls in the vase at noon is A \ C and B \ C,
resp. Thus, if A \ C is not empty, but B \ C is empty, then A must
contain numbers not in B.

Where do you disagree with this analysis?

>>
>> 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.

Whatever you were using, it isn't math.

--
"Reality is that I've worked all over the United States in so many
different jobs that I jokingly call myself The Pretender. I've been a
bartender, worked at Equifax, sold new cars for Honda, and that's from
about six months of my life." -- James S. Harris: a great Pretender.
.

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)