An uncountable countable set
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From: Virgil on 2 Nov 2006 00:58 In article <45497880$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > I've pointed out a couple contradictions in your position. TO has not pointed out any contradictions in any positions, except possibly his own. > The set > contains more and more elements at every time before noon (or at least > not less than at any time before 11:59), such that lim(t->0: balls)=oo. That " lim(t->0: balls)=oo" is nonsense. One could say that lim(t -> 0- : ......) might mean something but t -> 0, without modification, always means the two sided limit, and not even TO would be stupid enough to argue that. Or would he? Also what does "balls" mean here. If it is supposed to mean the cardinality of the set of balls, expressed as a function of time, it must say so, and if it means anything else it is nonsense. > You cannot empty the vase at any time before noon, and the vase only > empties through removal of balls, and that does not occur at noon, so it > can't happen before or at noon. It has happened at noon, whatever TO's objections may be since at noon every ball has already been removed. > If it doesn't happen at time t, or at > any time s<t, then it hasn't happened BY time t. By that argument the function f(t) = t cannot be 0 at t = 0, because it cannot occur before 0 and nothing can occur at 0. So TO strikes out again.
From: David Marcus on 2 Nov 2006 00:59 Tony Orlow wrote: > The iterations of insertion and removal are specified as such, and their > times specified, such that the number of balls is a function of time, > discontinuous for finite n or t, but constant for n, and constantly > exponential for t. > > It is true that: > 1) the vase contains balls, is thus non-empty, at every time before noon. > 2) No removals occur at noon. > 3) The vase can only become empty, after having contained balls, though > removal of balls. From 1-3, you conclude that the vase is non-empty at noon. Let's try this same logic on the modified problem where all the balls are inserted at one minute before noon. You've said that for this modified problem, the vase is empty at noon. However, it is true that 1) the vase contains balls, is thus non-empty, at every time before noon (we are only considering times that are after one minute before noon). 2) No removals occur at noon. 3) The vase can only become empty, after having contained balls, though removal of balls. Therefore, using your logic, the vase in the modified problem must be non-empty at noon. Yet, you conclude it is empty. Are you using logic or assuming your conclusion? -- David Marcus
From: Virgil on 2 Nov 2006 01:04 In article <eibv5p$o2b$1(a)news.msu.edu>, stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: > > Mike Kelly wrote: > >> > >> It logically precludes that balls without a finite natural number on > >> them get added to the vase, but that doesn't seem to bother you. Ho > >> hum. > >> > >> <snip more stuff about original experiment> > >> > > > The iterations of insertion and removal are specified as such, and their > > times specified, such that the number of balls is a function of time, > > discontinuous for finite n or t, but constant for n, and constantly > > exponential for t. > > > It is true that: > > 1) the vase contains balls, is thus non-empty, at every time before noon. > > 2) No removals occur at noon. > > 3) The vase can only become empty, after having contained balls, though > > removal of balls. > > > It is true that: > 1) the vase contains a finite number of balls at every time > before noon. > 2) No insertions occur at noon. > 3) The number of balls in the vase can only become infinite > through the insertion of balls > > Stephen God point! So it TO continues to insist that the vase contains infinitely many balls at noon, he must have sneaked them in himself outside of the rules of the gedankenexperiment.
From: imaginatorium on 2 Nov 2006 02:08 cbrown(a)cbrownsystems.com wrote: > imaginatorium(a)despammed.com wrote: > > Mike Kelly wrote: > > > Tony Orlow wrote: > > > > Mike Kelly wrote: > > > > > Tony Orlow wrote: > > > > <snip> > > > > > > > 2) How come noon "exists" in this experiment but it didn't exist in the > > > > > original experiment? Or did you give up on claiming noon doesn't > > > > > "exist"? What does that mean, anyway? > > > > > > > > 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. > > > > Hmm. Yes, there is no ball whose insertion time or removal time is > > noon. But it seems to me that this "happen" is underdefined in a way > > that can cause confusion. Does something "happen" to either of these > > functions at x=0: > > > > f(x) = 1 if x<0 ; 0 if x>=0 > > > > g(x) = 1 if x<=0 ; 0 if x>0 > > > > It seems to me that it is true (within the accuracy of normal > > communication) to say that both f() and g() "drop from 1 to 0 at x=0" > > even though the functions are different. > > > > Similarly, it seems to me that clearly something "happens" (in any > > normal sense) at noon in the standard vase problem - what happens is > > that the frenzy of unending sequences of insertion and removal come to > > a halt. > > > > And it follows by TO's unspoken assumptions that if something happens > at noon, then there is some other thing that /caused/ it to happen at > noon. But since nothing specified in the problem statement happens /at/ > noon which causes the frenzy to stop (it simply mysteriously stops) we > come to the conclusion: noon is a time when things happen without > cause. Yeah, sure - Tony's reasoning is pretty odd. I'm reminded somehow of various bits of argument I've read by Legal or Financial Experts. > Which is an absurd thing to say about a time; so either noon cannot > properly be said to be an actual time at all ("noon doesn't > exist/occur/happen"); or else something not specified in the problem > actually does happen at noon (such as the removal/addition of an > infinite number of infinitely labelled balls); or else the stopping of > the frenzy and its cause both occur at a time which is strictly between > all times before noon and noon itself (in which case, nothing happens > at all /at/ noon; instead, something happens at a time which is > indistinguishable from, but not the same as, noon). > > This is where/how Tony leaves the rails. > > The examples you give above of f(x) and g(x) are irrelevant; because > there is no specified "physical action" (ball removal or insertion) in > those examples; so the problem of "happenings" and "causes" is not an > issue; f and g are simply distractions from the original problem. You might be missing my point slightly. I'm trying to point out that the distinction between f() and g() is not one that could have a physical interpretation. In a sense the mathematics overspecifies, meaning that the use of functions over the reals is actually not quite sufficiently abstracted. > On the other hand, we can say that f(x) "correctly captures" the > removal of a single ball /at/ time 0, whereas g(x) can't capture any > such a thing; the ball would have to be removed /at/ some time strictly > between all times after 0, and 0 itself (although on reflection, this > may or may not be possible in Tony's worldview, which is hardly > consistent). Something is fishy here. You say f() correctly captures the "removal _at_ time 0", because f(0) is the value after the change. But consider a Feynmann space-time diagram (you know, the thing where an electron colliding with an antielectron, preceded by that antielectron having been spontaneously generated together with a(nother?) electron, can be viewed as a single electron travelling through a path so zigzag it goes backwards in time * details need correction). Suppose something "happens" to the thing going backwards in time: does this require function g() this time? Your interpretation seems to imply that if for example a ball is put in a vase from two o'clock to half past four, the corresponding interval for its presence is [2, 4.5). I don't see any reasonable basis for this claim. Of course you understand I'm not really disagreeing with you here - but I am insisting that the use of arguments such as "Nothing/Something _happens at_ noon" are likely to prolong the confusion. Brian Chandler http://imaginatorium.org
From: imaginatorium on 2 Nov 2006 02:51
Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Randy Poe wrote: <a lot of stuff> > >> I have seen and understood your argument. It "makes sense". It seems > >> logical. All balls are inserted and removed before noon, the same set, > >> it would seem. But the method of proof is not correct. > > > > Why? What are you basing this assertion on? That you don't agree with > > the conclusion? > > Yes. I am exploring exactly why. This is just another "la(rge)st finite" > argument. It doesn't "add up". Tony, many of us have heard this from you many times now, often accompanied by your mangled version of some mantra or other (I really mean a mantra, as in Om mani padme hum, not the popular meaning of "mantra"). I am correct in characterising this by saying that you do indeed accept and agree that there is no largest pofnat, no last natural number (under the normal definition), yet though you accpt it is true, you believe it is invalid to deduce anything from it. Is that really the case? Elsewhere, you just referred to " the twilight zone between zero and the smallest positive real". I suppose again you agree that there is no smallest positive real (because if q is the smallest positive real, then 0 < q/2 < q contradiction)? Yet you feel it makes sense to talk of a "zone" between one thing (0) and another thing that doesn't exist? Can you perhaps understand the difficulty most of us have with following your reasoning? Brian Chandler http://imaginatorium.org |