Re: An uncountable countable set


Tony Orlow wrote:
Mike Kelly wrote:
mueckenh@xxxxxxxxxxxxxxxxx wrote:
Han de Bruijn schrieb:

stephen@xxxxxxxxxx wrote:

Han.deBruijn@xxxxxxxxxxxxxx wrote:

Worse. I have fundamentally changed the mathematics. Such that it shall
no longer claim to have the "right" answer to an ill posed question.
Changed the mathematics? What does that mean?

The mathematics used in the balls and vase problem
is trivial. Each ball is put into the vase at a specific
time before noon, and each ball is removed from the vase at
a specific time before noon. Pick any arbitrary ball,
and we know exactly when it was added, and exactly when it
was removed, and every ball is removed.

Consider this rephrasing of the question:

you have a set of n balls labelled 0...n-1.

ball #m is added to the vase at time 1/2^(m/10) minutes
before noon.

ball #m is removed from the vase at time 1/2^m minutes
before noon.

how many balls are in the vase at noon?

What does your "mathematics" say the answer to this
question is, in the "limit" as n approaches infinity?
My mathematics says that it is an ill-posed question. And it doesn't
give an answer to ill-posed questions.
You are right, but the illness does not begin with the vase, it beginns
already with the assumption that meaningful results could be obtained
under the premise that infinie sets like |N did actually exist.

The meaningful result is that if you allow "|N exists" then the vase
empties at noon. Even if you don't allow that in your mathematics, you
can surely accept the logical conclusion that IF you allow that THEN
the vase is empty at noon. No?

Only if you change the order of events,

In the original problem, every ball that is added is added at a time
before noon and has a removal time before noon. To note this isn't
changing the order of events. It's just to point out the utter
absurdity of your position.

For ball n, as stated in the original problem :

Insertion time = -1/(2^(floor((n-1)/10)))
Removal time = -1/(2^(n-1))

or refuse to say when the vase
empties or how. Any "|N" aside, the problem clearly states that ten
balls are added and then one removed, per iteration, so if the vase
emptied, it could only be with the removal of that 1 ball, not with the
addition of the ten balls, since that would require that there had been
-10 balls in the vase. But, for there to be 1 ball left, which when
removed left an empty vase, ten would have been inserted right
beforehand, meaning there had to have been -9 balls in the vase. Neither
negative count is possible, therefore the vase could not have emptied.

There is no last step in an infinite sequence of events. You still
don't understand this a year later. Pathetic, really.

The first point in time after the experiment begins at which the vase
is empty is noon. There is no iteration at which the vase becomes
empty. The vase does not become empty at an iteration. It becomes empty
at noon and not before.Every iteration is a time before noon. There is
no last iteration before noon.

When you come to two logical conclusions given two lines of thought, how
do you resolve that?

By noting that your conclusion is not logical; it is a kludge based on
your errant intuition and vomiting vacant verbiage that you think
sounds mathematical. You think to maniuplate limits in your argumentst.
But how do you know when you are using them in a valid way? You have
*no idea* as all you know about limits is gleaned from reading this
newgroup and skimming online sources like Wikipedia and Mathworld.

lim(n->oo,S(T(n))) = S(lim(n->oo),T(n)) is true when S is continuous

Did you know this? I highly doubt it. So how can anyone expect to trust
your arguments based on limits, if you don't know when it's valid to
apply them?

You don't do actual mathematics, you just like to waffle about vague
ideas you have based on your misconceptions of what mathematics is.
This isn't going to be productive.

Unless I am very much mistaken, a large part of the whole *point* of
the ball and vase problem is to demonstrate that one cannot always mess
around with limits willy-nilly and expect to obtain valid results. Of
course, many will stumble when first seeing it because they are used to
informal manipulation of limits to obtain what feels like the "right"
result. It's supposed to be a prod to make you realise that it's
important to be very careful that any limit manipulation you do is
based on a rigorous bedrock. But you don't even know what the rigorous
rules governing when limits are a valid tool *are*.

I'm expecting you to respond with a comment that you have your own
version of limits and you can use them as a magic wand to deduce
whatever your intuition tells you is the "right answer". But you don't
have a mathematical tool. You have your vague ideas which are backed up
by nothing more than "This feels right to Tony Orlow".

You're attacking the problem with a blamanche when what you need are
precision tools.

--
mike.

.

Relevant Pages

  • Re: An uncountable countable set
    ... The mathematics used in the balls and vase problem ... Each ball is put into the vase at a specific ... Only if you change the order of events, or refuse to say when the vase empties or how. ...
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  • Re: An uncountable countable set
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  • Re: infinity
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