Re: Cantor Confusion

*** T. Winter wrote:
In article <1160302222.613036.300930@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> Han.deBruijn@xxxxxxxxxxxxxx writes:

> If it is so simple, where then come these heated debates (about the
> Balls in a Vase at noon) come from?

Because some of the people in the discussion only use intuition, and not
provable concept, but they keep stating that there intuition tells them
that there are contradictions, without giving proof of any of this.


I agree.

If you are, say, discussing in a context where the axiom of infinity is
assumed, you can not get a contradiction when during your proof you, at
one stage or another, use the contradiction of that axiom.


I agree again!

The balls in vase problem suffers because the problem is not well-defined.
Most people in the discussion assume some implicit definitions, well that
does not work as other people assume other definitions. How do you
*define* the number of balls at noon?

I disagree.

This is no more difficult than asking "how do you *define* the number
of balls at pi/10 seconds before midnight?"

To ask "how many balls are in vase A at time t?" is equivalent to
asking "what is the cardinality of the set of balls in vase A at time
t?"

As the question is phrased, we can define a function f of the
(continuous) time t yielding a well-defined set of naturals: f(t) = {n
in N : 1/(10*n) < -t < 1/n}. This is consistent with, e.g., f(-9/10) =
f(-1); and to my mind, "n in f(t)" is the only sensible interpretation
to the question "Is ball n in vase A at time t?"

We /then/ define a function numBalls(t) on these sets, equivalent to
the cardinality of f(t). And so we can ask "what is the number of balls
at time t?", and always get a well-defined (and finite) value.

It is fallacious to base one's argument on the assumption that the
problem is /equivalent/ to defining a function g on N ("the number of
balls at step n"), satisfying g(n)*9 = g(n+1), and that therefore
either (a) t = 0 implies n = oo, and that therefore at time t, f(n) =
f(oo) = oo, or (b) since n is always finite in the problem, "noon never
occurs".

This approach misses the point. /Time/ is the /independent/ variable
here, not n.

The question isn't "Based on the step B in the process which is
equivalent to noon, how many balls are there at step B?", which would
indeed be a limit problem; the question is, "Given that we can define
the function f(t), what is |f(0)|?"

You can not use limits, because the
limit does not exist when you use standard mathematics.

And in fact this is not a problem of limits.

Cheers - Chas

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