Re: Cantor Confusion


mueckenh@xxxxxxxxxxxxxxxxx wrote:
William Hughes schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
Alan Morgan schrieb:


As I have inductively gone through the entire list of balls introduced
into the vase and found that each of them has been removed before noon,
why should stating that trivial fact be considered a joke?

But you cannot go inductively through the cardinal numbers of the sets
of balls in the vase? They are 9, 18, 27, ..., and, above all, we can
show inductively, that this function can never decrease.

You think that's bad? I have an even simpler situation! Add one ball
at 1 minute to noon, another ball at half a minute to noon, another
at 1/4 minute to noon, and so on. The number of balls in the vase before
noon is always finite, but somehow, miraculously, at noon the number of
balls in the vase becomes infinite. When, oh when, does that transition
from finite to infinite happen?

I submit that this is just as wierd a result as the original problem.

Weird is that adding 9 balls instead of 1 per transaction leads to zero
balls.
Weird is that taking off 1 ball per transaction leads to all balls
taken off and no ball remaining, if the enumeration is 1,2,3,... but to
infinitely many balls remaining, if the enumeration is 10, 20, 30, .. .
This in particular is weird because there is a simple bijection between
1,2,3,... and 10, 20, 30, ...


Right. No matter which balls you pick you are going to remove an
infinite
number of balls. So the number of balls you remove does not matter.
Which balls you remove does matter.

Fine. The question is, however, is this set "all balls" or how many
balls remain in the vase at noon?

As there is more than one way to remove an infinite set of balls
(remove all balls, starting with 1, remove just the even balls,
remove just the primes, remove just the multiples of 10 ...) the
answer to your question depends on exaclty which balls are
removed. So the set removed may or may not be the
set "all balls".

- William Hughes

.

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