Re: An uncountable countable set

David Marcus wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
David Marcus wrote:
Your assumptions seem consistent with the following formulation of the
problem.

For n = 1,2,..., define

A_n = -1/floor((n+9)/10),
R_n = -1/n.

For n = 1,2,..., define a function B_n by

B_n(t) = 1 if A_n < t < R_n,
0 if t < A_n or t > R_n,
undefined if t = A_n or t = R_n.

Let V(t) = sum{n=1}^infty B_n(t). What is V(0)?
Yes; and in fact I chose (1)-(8) to be consistent with just about any
sensible interpretation of the problem as given.

But I'm currently trying with Tony to completely avoid numerical
arguments such as the above, which rely on a complicated definition of
"the number of balls at time t", in favor of the much simpler to agree
with statement "either there is a ball in the vase at time t, or the
number of balls in the vase at time t is 0; and not both".

It is worth a try. Although, I think you'd do better to limit your replies to a few connected paragraphs rather than reply to so many individual paragraphs of Tony's post. While carrying on many threads of a discussion in one post saves time (like I'm doing in this post), it only seems to work if both people are basically on the same wavelength. Otherwise, people pick and choose which comments they reply to.

I mean, given his confusion over simple logical arguments like "If
(A->not A), then not A", I shudder to think what subtle
misunderstandings exist in his version of "define a sequence of
functions indexed by n in N".

Perhaps. Although, I think he might have taken Calculus at some point. Many people who take Calculus learn something about functions despite being unable to reason logically.

It is rather amazing.
It's also sort of fascinating - how can one /not/ understand the
argument, and yet give the impression of understanding /some/ sort of
logic? It's like some sort of mental blind spot.

Do you really think he understands any logic? I believe that 98% or more of people don't think conceptually/logically. Instead they rely on the brain's amazing ability to do pattern matching. Pattern matching is extrememly useful, but it dosn't do logic.

The logic seems to be that the limit of the number
of balls in the vase as we approach noon is infinity, so the number of
balls in the vase at noon must be infinity, but all numbered balls have
been removed, therefore the infinity of balls in the vase at noon aren't
numbered. It does have a sort of surreal appeal.
If we assume at the start that the number of balls at t=0 is /anything
but/ 0 (as TO apparantly does, although he has yet to realize it), then
pretty much anything goes. Let your imagination roam! There are a prime
number of cubical balls in the vase at noon! ZFC is inconsistent!
Cantor is alive and living in Brooklyn New York! I am the current King
of France!

I don't agree that he is assuming that. I think he isn't reasoning logically at all. The number of balls approaches infinity as time approaches noon. If you imagine a vase filling up with an infinite number of balls, it is rather hard to imagine them suddenly all disappearing. Of course, mathematics isn't constrained by our imagination. It relies on precise definitions and logic. And, functions do not have to be continuous.

So, David, you think the fact that balls leave the vase only by being removed one at a time, and the fact that at all times before noon there are balls in the vase, and the fact that at noon there are no balls in the vase, is consistent with the fact that no balls are removed at noon? How can you not see the logical inconsistency of an infinitude of balls disappearing, not just in a moment, but at no possible moment? Are you so steeped in set theory that you cannot see that an unending sequence of +10-1 amounts to an unending series of +9's which diverges? What is illogical about that?

In your set-theoretic interpretation of the experiment there is a problem which makes your conclusion incompatible with conclusions drawn from infinite series, and other basic logical approaches. It is not that I don't understand how your logic works. It's that I see clearly that it doesn't, and I'm trying to precisely pin down exactly where the error is. It's not an easy task, since this transfinite theory is rather well crafted and tweaked over the years. However, there are clear reasons, once the matter is fully investigated, why the logic fails. The conclusion produces clear contradictions in terms of a time of emptying and the requirement at some point of a negative number of balls in the vase in order for it to empty at all, and it all derives from using the Zeno schedule to complete a sequence which has no end, hiding this fact in a time singularity at t=0.

Very basic logic would hold that, if the vase is not empty at any time t such that -1<=t<0, and the vase is empty at t=0, then balls were removed at t=0, since that's the only way the vase can become empty. However, t=0 corresponds, according to the stated schedule, to infinite index n in the sequence, and an infinite label on a ball, which is not allowed, as per the experiment. Therefore, no ball can be removed at t=0, and the vase cannot become empty at that point, or at any point before.

I asked you when you thought the vase became empty. You avoided the question, saying it was interesting, and then going on with your same tired formulation of the problem, as if I haven't followed the logic and pointed out the flaw in the approach.

So, answer the question. When does this miracle of emptiness occur?

Tony
.

Relevant Pages

  • Re: Questions about the marble problem.
    ... >>>definition of the number of balls in the urn at time s to infinity by ... namely the set of marbles in the vase at that time. ... >> of sets with the property that the cardinality of the set ...
    (sci.math)
  • 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: Infinity : An interesting variant
    ... This is a succinct demonstration of fighting fire with fire ie of infinity ... to vase B, but the inclusion of ball zero would exceed your neat figure of 1 ... (try recasting the model with extra balls). ...
    (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)
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