An uncountable countable set
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From: David Marcus on 2 Nov 2006 19:21 MoeBlee wrote: > Lester Zick wrote: > > On 2 Nov 2006 11:21:10 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > > >> I'm perfectly content to stay right here while I refine my definitions > > >> and terminology to bring them into better conformance with standard > > >> mathematical usage and more appropriate neomathematical forensic > > >> modalities. > > > > Evasion noted. > > You just quoted yourself, then commented 'Evasion noted'. Nicely done. Even Lester has to be right occasionally. -- David Marcus
From: Virgil on 2 Nov 2006 19:58 In article <MPG.1fb454701c352cd998980f(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > cbrown(a)cbrownsystems.com wrote: > > (Side bar: I've always been slightly tweaked by the events at time t = > > -1. Ball 1 is simultaneously added to and removed from the vase. Is > > Ball 1 ever actually "in" the vase? What if we had said "at time t = > > 1/n, remove ball n and add balls (10*(n-1)+1) to 10*n inclusive?" Would > > we actually know that Ball 1 isn't /still/ in the vase? In which case, > > we've been wrong all along - the vase is /not/ empty at noon: Ball 1 is > > in the vase at noon!). > > I've also found that curious. If we just blindly translate the words > into math, we get that ball 1 is in the vase at time t iff -1 <= t < -1. > So, ball 1 is never in the vase. Wish is why most of us carefully avoid saying what is in the vase at any transition instants. We have no trouble with a ball being in the vase at times strictly between the time of insertion and time of removal, but determining where it is while it is, at least metaphorically, "in motion" is a tough question.
From: cbrown on 2 Nov 2006 20:24 David Marcus wrote: > cbrown(a)cbrownsystems.com wrote: > > David Marcus wrote: > > > cbrown(a)cbrownsystems.com wrote: > > > > imaginatorium(a)despammed.com wrote: > > > > > > > > You say f() correctly captures the "removal > > > > > _at_ time 0", because f(0) is the value after the change. > > > > > > > > Right - because the event which we call "the vase becomes empty at 0" > > > > must be "preceded" by some event which /causes/ that "becoming" > > > > (regardless of rest frame). In this case, BTW, "a precedes b" means > > > > "the time at which a occurs <= the time at which b occurs, for evey > > > > rest frame". Although giving that away is really a form of cheating. > > > > > > > > Under these "obvious" constraints, f() describes a certain real world > > > > situation; if the 'cause' of f(0) = 0 is that the indicator function > > > > k(t) = 1 iff a ball is removed at time t=0 has k(0)=1 . The same, sadly > > > > or happily, cannot be said of h(). > > > > > > What's with the empty parentheses? The names of the functions are "f" > > > and "h", not "f()" and "h()". > > > > Hey! Don't blame me! He started it! > > I wasn't blaming you! Although, just because he starts it, doesn't mean > you have to keep doing it! > I only hope he doesn't jump off the Empire State Building. Cheers - Chas
From: cbrown on 2 Nov 2006 20:28 David Marcus wrote: > cbrown(a)cbrownsystems.com wrote: > > (Side bar: I've always been slightly tweaked by the events at time t = > > -1. Ball 1 is simultaneously added to and removed from the vase. Is > > Ball 1 ever actually "in" the vase? What if we had said "at time t = > > 1/n, remove ball n and add balls (10*(n-1)+1) to 10*n inclusive?" Would > > we actually know that Ball 1 isn't /still/ in the vase? In which case, > > we've been wrong all along - the vase is /not/ empty at noon: Ball 1 is > > in the vase at noon!). > > I've also found that curious. If we just blindly translate the words > into math, we get that ball 1 is in the vase at time t iff -1 <= t < -1. > So, ball 1 is never in the vase. > Well, hardly "blindly"; we need to appeal to (rather plain) assumptions such as my (1)..(4) in order to make this claim. It does speak to the acceptance of "ball in the vase from 2 'til 4" as meaning the ball is in the vase during the half open interval [2,4). Cheers - Chas
From: Tony Orlow on 2 Nov 2006 22:36
stephen(a)nomail.com wrote: > Randy Poe <poespam-trap(a)yahoo.com> wrote: > >> Mike Kelly wrote: >>> Tony Orlow wrote: >>>> Mike Kelly wrote: >>>> Nothing is allowed to happen at noon in either experiment. >>> Nothing "happens" at noon? I take this to mean that there is no >>> insertion or removal of balls at noon, yes? Well, I agree with that. >>> But what relevence does this have to the statement "noon does not >>> exist"? What does that even *mean*? >>> >>> When you've been saying "noon doesn't exist", you actually mean to say >>> "no insertion or removal of balls occurs at noon"? >>> >>> How about this experiment, does noon "exist" in this experiment : >>> >>> Insert a ball labelled "1" into the vase at one minute to noon. >>> >>> ? > >> I think that when Tony and Han say "noon doesn't exist" they >> really mean "there is no noon on the clock in that experiment", >> as a way of saying "I have no idea how to answer questions about >> noon in that experiment, so I'll say that there is no noon and that >> way I don't have to answer any such questions." Or, we say that introducing noon into the situation as the time of an event creates a contradiction, since 1/n cannot be zero for any natural n. Since the vase can only become empty (correct me if I'm wrong) if balls are removed, and no balls are removed at noon, it cannot become empty at noon. On the other hand, at every finite time -t before noon=0, there are a finite but exponentially growing number of balls in the vase, with respect to t. That is of course because n, the count of iterations of adding 10 and removing 1, has an exponential relationship with respect to t (well, using 1/2^n anyway), and a linear relationship to the "size of the set". With each new iteration n, 9 balls are added. Now, I understand you may not agree with the idea that the size of a set might "change", but if you are talking about sequences, either over t or over n, then you are not talking about static sets, but about progressions. You are getting into measure, but cardinality affords no measure except in the finite case, where it's unavoidable. Basically, the n and the t are buried in the noon. > That sounds about right. It is also interesting that for some > reason "noon" is not a necessary part of the problem, but "rates" > and "iterations" are. Any formulation that ignores rates is > incorrect, but it is okay to ignore "noon", despite the fact > the actual question is about "noon". > > Stephen > It's about the pertinent variables not being lost. Tony |