Re: infinity

Stephen wrote:

<a bunch of good stuff to Tony>

This post of Stephen's has got to be the most ironic thing yet to
appear on this thread. I'm sure it is entirely unintentional.

First, a paraphrase:

> 'After an infinite number of steps' is not very
> precise, and does not describe how you get from a [positive]
> number to [0]. There either is a step
> that starts with a [positive] number of balls and ends with
> [0] balls, or there is not a step that
> starts with a [positive] number of balls and ends with [0]
> balls.
[...]

> You are still talking about something that does not happen.
> There is no such step. Why would doing something infinitely
> many times make the nonexistent step suddenly happen?
> Why does it [...] cause the nonexistent step of there
> being 0 balls to happen?
[...]

> And you are adding balls a [positive] quantity at a time and claiming
> that the quantity becomes [0]. So in order to get an [0]
> number, there must be a step where n was [positive], and then n+9 was
> [0].


And then comes this gem:

> You still have not explained
> why you get to have results that are not produced by any
> step, but noone else is.

This is not quite correct - nearly EVERYONE on this thread claims such
results, but no one has offered such an explanation.


A casual perusal of the various recent "Cantor" threads reveals the
folly of considering the situation after an infinite process has
"completed". The likes of Mueckenheim and Orlow are constantly
reminded that an infinite process /does not/ terminate and that it is
/meaningless/ to talk about the end of such a process. The only way
one can describe and "end result" is in terms of limits. Otherwise we
would have a smallest positive real, 0.999... would be < 1 (since we
wouldn't need limits to evaluate the former), the Cantor "antidiagonal"
would end at either the bottom, side, or corner of a rectangle, and we
could find two rational numbers without another rational in between
them.

The very idea that each ball added is removeable at some later time
depends on the idea that an infinite process *does* *not* *end*.

In most threads, posters are very clear about this. This thread is a
curious exception (as were its prior incarnations - this problem seems
to come up here regularly). Elsewhere, posters rightly object to
Orlow's "construction" of the naturals done by adding 1 to a variable
an infinite number of times. Here, some of the same people have no
qualms about considering an initially empty vase "after" one has
sequentially added balls 1, 2, 3, ... to it (adding ball n at 1/n
minute before noon).


According to the problem statement, the only things that are done to
change the state of the vase are done at times of the form (noon -
1/n) where n is a natural number. Let t=0 represent noon, so we can
write this as t = -1/n. Nothing is done at noon to change the state
of the vase. Nothing "outside" the process can affect the vase. If
the vase becomes empty it does so as a consequence of the process
itself. The process is nothing more than the aggregation of its
parts, the individual steps, so everything that happens happens as a
result of one of those steps. The vase does not become empty "after"
the process "completes" (as if that were meaningful), because nothing
happens to it then. For those who claim that the vase is empty at
noon, at what precise time did the vase become empty? It can't be
*at* noon, since nothing happens at that moment, so it must be at t =
-epsilon. Why are no balls added between then and noon? Is there no n
for which 1/n < noon - epsilon? Is epsilon infinitesimal? Note that,
for the vase to be affected at all, epsilon must = 1/n for some n,
which means the vase must be emptied at t = -1/n, but we know that at
that time a net of 9 balls are added. Every single operation on the
vase has the effect of adding balls to it. No other kind of operation
is ever performed on the vase.

More formally, for each t of the form -1/n, the number of balls in the
vase increases, so the vase cannot become empty at time t. For all
other t (again, this includes t=0), nothing about the vase changes, so
it cannot become empty at time t. Therefore the vase can never be
empty after the first ball has been inserted.

Combining this with the argument that the vase cannot contain any
particular ball at noon [most writers only consider this half of the
argument] yields a contradiction, so the vase cannot exist at noon.


The problem statement essentially defines a function from R to ordered
pairs of sets of naturals:

for x = -1/n, f(x) = ( {10n-9, 10n-8, ..., 10n}, {n} ) = ( I_x, O_x
)
for all other x, f(x) = ( {}, {} ) = ( I_x, O_x )

We could then try to define a function for the contents of the vase:
V(t) = U{x<t}I_x \ U{x<t}O_x, the set difference between a pair of
uncountable unions of sets. But are uncountable unions defined? It is
tempting to define f as a function on the naturals, to get countable
unions, but then it is blisteringly obvious that one cannot talk about
V(0), the situation at noon.

It's easy to see that each n is accounted for exactly once among the
I_x's and once among the O_x's. Thus, f(R-) = ( U{x<0}I_x, U{x<0}O_x
) = ( N, N ), so there's no problem with the static definition, as long
as the unions can be defined.

The difficulty arises when one tries to think of this as a /sequential
process/, where each value of x (or even just each n) is considered in
turn. One never reaches noon. "But we always reach noon", you say.
"Time waits for no [one]!" You're just being misled by your intuition
about physical reality. Mathematically, there's no reason that we
have to reach noon.

In some statements of the problem, the steps are to be performed one
minute apart, and it is clear that the process runs forever, that is,
without ending. Mapping the time line from [1,oo) to [-1,0) does not
change this. Asking about the state of the vase at noon is no more
meaningful than asking for a largest natural number.


Gordon
clavier at comcast dot net

.

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: An uncountable countable set
    ... above) and given that the "moment the vase becomes empty" means the ... Yes, now, when nothing occurs at noon, and no balls are removed, what ... The fact that you have no upper bound to the naturals. ...
    (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)
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