An uncountable countable set
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From: Tony Orlow on 26 Oct 2006 11:59 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> Virgil wrote: >>>> < endless reiterations of the following > >>>>> The only question is "According to the rules set up in the problem, is >>>>> each ball which is inserted into the vase before noon also removed from >>>>> the vase before noon?" >>>>> >>>>> An affirmative answer confirms that the vase is empty at noon. >>>>> A negative answer violates the conditions of the problem. >>>>> >>>>> Which answer does TO choose? >>>> God, are you a broken record, or what? Let's take this very slowly. Ready? >>>> >>>> Each ball inserted before noon is removed before noon, but at each time >>>> before noon when a ball is removed, 10 balls have been added, and 9/10 >>>> of the balls inserted remain. Therefore, at no time before noon is the >>>> vase empty. Agreed? >>>> >>>> Events including insertions and removals only occur at times t of the >>>> form t=-1/n, where n e N. Where noon means t=0, there is no t such that >>>> -1/n=0. Therefore, no insertions or removals can occur at noon. Agreed? >>>> >>>> Balls can only leave the vase by removal, each of which must occur at >>>> some t=-1/n. The vase can only become empty if balls leave. Therefore >>>> the vase cannot become empty at noon. Agreed? >>> Not so fast. What do "become empty" or "become empty at" mean? >> "Not so fast"???? We've been laboring this point endlessly. The vase >> goes from a state of balledness to a state of balllessness starting at >> time 0. > > Agreed. > >> Balls have to have been removed for this transition to occur. > > Yes, but they don't have to have been removed at time 0. In order for emptiness to occur at that time, removals have to occur at that time, if removals are what causes the emptiness. Was that too fast? > >>>> It is not empty, and it does not become empty, then it is still not >>>> empty. Agreed? >>>> >>>> When you bring t=0 into the experiment, if anything DOES occur at that >>>> moment, then the index n of any ball removed at that point must satisfy >>>> t=-1/n=0, which means that n must be infinite. So, if noon comes, you >>>> will have balls, but not finitely numbered balls. In this experiment, >>>> however, t=0 is excluded by the fact that n e N, so noon is implicitly >>>> impossible to begin with. >
From: Tony Orlow on 26 Oct 2006 12:31 RLG wrote: > "Tony Orlow" <tony(a)lightlink.com> wrote in message > news:453fac14(a)news2.lightlink.com... >> RLG wrote: >>> "Tony Orlow" <tony(a)lightlink.com> wrote in message >>> news:453e4a3f(a)news2.lightlink.com... >>> >>> From my reading of this issue the vase is empty at noon, as David Marcus >>> says. But Tony, I have a question for you. Suppose we put one more ball >>> into the vase, at any time before noon, and that ball is labeled "oo". >>> At exactly noon that ball is removed from the vase. At noon is the vase >>> empty or does it contain the ball labeled "oo"? >>> >>> -R >>> >> Hi RLG. Welcome to the conversation. >> >> According to the experiment, all balls inserted and removed are finite, so >> that doesn't really apply. Every ball n is inserted at time -1/n >> (or -1/2^n originally, but it doesn't matter), so ball oo cannot be >> inserted before noon. >> >> But, if you want to entertain the idea of inserting an extra ball named >> "oo", or "Bill", or "RLG", the addition of a ball is not going to make the >> vase any more empty. I wonder what logical implication you think that >> has.... > > My apologies if the logical implication was unclear. What I was trying to > get at was the issue of how well point set theory does or does not apply to > the continuum. Suppose a ball labeled "oo" is put into the vase five > minutes before noon. I asked that if at exactly noon the ball labeled "oo" > is removed from the vase is the vase empty at noon or does it contain the > ball labeled "oo"? Is the vase time for the "oo" ball [-5,0] or [-5,0) and > is there a difference between these two in terms of temporal duration? In > the [-5,0] case the ball is in the vase at the instant t=0 but not at any > time t>0. Since the ball is in the vase at t=0 it is not removed at that > time so it must be removed at some time t>0. But for every t>0 there is a > (t/2)>0 such that t>(t/2)>0 and so at no t>0 is the ball removed from the > vase. It seems that in the [-5,0] case there is no instant at which the > "oo" ball is removed from the vase. In the [-5,0) case the ball is not in > the vase at time t=0 so it seems it should have been removed at some time > t<0. But the "oo" ball is in the vase for all t<0 and so at no t<0 is the > "oo" ball removed from the vase. It seems that in the [-5,0) case there is > no instant at which the "oo" ball is removed from the vase. So I ask you > again, if the "oo" ball is removed from the vase at exactly noon, is it in > the vase at exactly noon or not? Maybe this is a joke question. > > Before considering that question too deeply, consider another question. The > length of the two intervals [0,1] and [0,1) is both 1. In point set theory > these are two different sets, the latter set is identical to the former set > except it does not contain the number 1. Yet, in terms of distance, these > distinctions make no sense. If you have a stick one meter long and take > nothing at all off its right end, you haven't changed anything. In terms of > distance and time distinctions like [-5,0], [-5,0), [0,1], and [0,1) are > meaningless. This raises the question as to how well point set theory > applies to the continuum and I remember reading somewhere that Godel had > concerns about this. In my opinion, point set theory is the best tool > presently available for studying the continuum but I recognize that there > are some limitations to it. I share your and Godel's concerns about point set theory, or at least have some concerns of my own, which arose in a conversation regarding the diagonal from (0,0) to (1,1) as the limit of a staircase between those two points, as the number of steps approached oo, since the points between the two objects become arbitrarily close. This was Chas' counterexample to my claim that inductive proof can be extended to the infinite case, with certain precautions. His claim was that, if it were true, then the diagonal should have length 2, as do all the finite staircases. My counterclaim was that the staircase in the limit was not the same object as the diagonal, but was a fractal line, maintaining its right angle on an infinitesimal scale. To demonstrate, I concocted a segment sequence topology which clearly showed the difference between the two. So, yes, I agree there are issues with point set approaches when it comes to measure. You mention measure above, in terms of [0,1] and [0,1) having the same measure, but being different sets. I actually consider the second to have infinitesimally smaller measure than the first, a difference of the unit infinitesimal. Applying this notion to the experiment under discussion, I would say the experiment occurs in [-1,0) and that 0 is not included, or else n must become infinite, which is not allowed. If one is to entertain the notion of noon in the experiment, and claim the vase is empty at noon, it had to have become empty, either before noon, or at noon. When it comes to the question of whether a ball is in or out at its moment of removal, it's both for that instant. We can agree to a convention that addition or removal affects the vase from that moment forward, or for all subsequent moments. I don't think it makes a difference. The event still occurred at that moment, right? > > This whole issue that you and other have been arguing is called the > Ross-Littlewood paradox. I suggest you read about it at > http://en.wikipedia.org/wiki/Supertask. Wikipedia phrases it this way: > > "Suppose you had a jar capable of containing infinitely many marbles, and an > infinite collection of marbles labeled 1, 2, 3, and so on. At t=0, marbles 1 > to 10 are placed in the jar, at t=1/2 11 to 20 are placed in the jar but > marble 1 is taken out. At t=3/4 marbles 21 to 30 are put in the jar and > marble 2 is taken out: in general at time t=1-(1/2)^n, the marbles (10*n + > 1) to (10*n + 10) are placed in the jar and marble n is taken out. The > question is: How many marbles are in the jar at t=1?" > > Here is my take on the whole Ross-Littlewood issue. If all the marbles are > labeled as described above, they should all be out of the jar at the end of > the supertask. So I agree with your antagonists on this point. But if they > were labeled differently that need not be the case. Suppose at t=0 ten > ma
From: Mike Kelly on 26 Oct 2006 12:38 Tony Orlow wrote: > Mike Kelly wrote: > > Randy Poe wrote: > >> Tony Orlow wrote: > >>> Virgil wrote: > >>>> In article <453e4a85(a)news2.lightlink.com>, > >>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> > >>>>> David Marcus wrote: > >>>>>> Tony Orlow wrote: > >>>>>>> David Marcus wrote: > >>>>>>>> Tony Orlow wrote: > >>>>>>>>> David Marcus wrote: > >>>>>>>>>> Tony Orlow wrote: > >>>>>>>>>>> Your examples of the circle and rectangle are good. Neither has a > >>>>>>>>>>> height > >>>>>>>>>>> outside of its x range. The height of the circle is 0 at x=-1 and x=1, > >>>>>>>>>>> because the circle actually exists there. To ask about its height at > >>>>>>>>>>> x=9 > >>>>>>>>>>> is like asking how the air quality was on the 85th floor of the World > >>>>>>>>>>> Trade Center yesterday. Similarly, it makes little sense to ask what > >>>>>>>>>>> happens at noon. There is no vase at noon. > >>>>>>>>>> Do you really mean to say that there is no vase at noon or do you mean > >>>>>>>>>> to say that the vase is not empty at noon? > >>>>>>>>> If noon exists at all, the vase is not empty. All finite naturals will > >>>>>>>>> have been removed, but an infinite number of infinitely-numbered balls > >>>>>>>>> will remain. > >>>>>>>> "If noon exists at all"? How do we decide? > >>>>>>>> > >>>>>>> We decide on the basis of whether 1/n=0. Is that possible for n in N? > >>>>>>> Hmmmm......nope. > >>>>>> So, noon doesn't exist. And, there is no vase at noon. I thought you > >>>>>> were saying the vase contains an infinite number of balls at noon. > >>>>>> > >>>>> If the vase exists at noon, then it has an uncountable number of balls > >>>>> labeled with infinite values. But, no infinite values are allowed i the > >>>>> experiment, so this cannot happen, and noon is excluded. > >>>> So did the North Koreans nuke the vase before noon? > >>>> > >>>> The only relevant issue is whether according to the rules set up in the > >>>> problem, is each ball inserted before noon also removed before noon?" > >>>> > >>>> An affirmative confirms that the vase is empty at noon. > >>>> A negative directly violates the conditions of the problem. > >>>> > >>>> How does TO answer? > >>> You can repeat the same inane nonsense 25 more times, if you want. I > >>> already answered the question. It's not my problem that you can't > >>> understand it. > >> Your response requires that the vase contains balls which were > >> never, by the stated rules, put in. > >> > >> You keep saying things like "if the clock runs till noon there are > >> balls with infinite numbers on them" even though the rules say there > >> are > >> no balls with infinite numbers on them. How do you reconcile that? > >> > >> If I put in balls 1, 2, 3 and stop, can the clock tick till noon > >> without > >> requiring a 4th ball? > >> > >> If I specify times for balls 1-1000 only, can the clock till noon > >> without > >> requiring a 1001-th ball? > >> > >> How is it, in your world, that when I specify times for all natural > >> numbered > >> balls, I am required to put in balls that don't have natural numbers? > > > > The problem is that Tony thinks time is a function of the number of > > insertions you've gone through. In order to "get to" any particular > > time you have to perform the insertions "up to" that point. He then > > thinks that if you want to "get to" noon, you have to have performed > > some "infinite" (whatever that means) iterations, where balls without > > natural numbers are inserted. That this is obviously not what the > > problem statement says doesn't seem to bother him. Nor that it's > > absolutely nothing like an intuitive picture of what time is. > > Time is ultimately irrelevant in this gedanken, but if it is to be > considered, the constraints regarding time cannot be ignored. Is time relevent to the question or isn't it? If it isn't, why must these "constraints" be respected? > Events occurring in time must occupy at least one moment. I have no idea what this is supposed to mean. > > Obviously, time is an independent variable in this experiment and the > > insertion or removal or location of balls is a function of time. That's > > what the problem statement says: we have this thing called "time" which > > is a real number and it "goes from" before noon to after noon and, at > > certain specified times, things happen. There are only > > naturally-numbered balls inserted and removed, always before noon. > > Every ball is removed before noon. Therefore, the vase is empty. > > No, you have the concept of the independent variable bent. No, you have the concept of the independent variable bent. > The number of balls is related to the time by a formula which works in both directions. For any iteration in the sequence of insertions/removals you can work out what time it occurs at if you know what number the iteration is indexed by. This doesn't imply that noon "does not exist" unless there is an iteration that corresponds to it. That is a complete non sequitur and, I think, the root logical error that you make. > So, when does the vase become empty? Nothing can occur at noon, as far > as ball removals. AT every time before noon, balls are in the vase. So, > when does the vase become empty At every time before noon (after 1 minute to noon) there are balls in the vase. At noon, there are no balls in the vase. So I guess one would say the vase "becomes" empty at noon. >, and how? By every ball that was inserted having been removed. Now correct me if I'm wrong, but I think you agreed that every "specific" ball has been removed before noon. And indeed the problem statement doesn't mention any "non-specific" balls, so it seems that the vase must be empty. However, you believe that in order to "reach noon" one must have iterations where "non specific" balls without natural numbers are inserted into the vase and thus, if the problem makes sense and "noon" is meaningful, the vase is non-empty at noon. Is this a fair summary of your position? If so, I'd like to make clear that I have no idea in the world why you hold such a notion. It seems utterly illogical to me and it baffles me why you hold to it so doggedly. So, I'd like to try and understand why you think that it is the case. If you can explain it cogently, maybe I'll be convinced that you make sense. And maybe if you can't explain, you'll ad
From: Randy Poe on 26 Oct 2006 13:12 Tony Orlow wrote: > t=0 is precluded by n e N and t(n) = -1/n. Really? I hope you will accept as true that noon occurred yesterday. Let's define noon yesterday as t=0. Now let's define a set of values t_n = -1/n seconds for n=1, 2, 3, ... , that is, for all FINITE natural numbers n. Has my giving these names to those times somehow precluded noon yesterday from occurring? Retroactively? - Randy
From: stephen on 26 Oct 2006 13:31
Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>> imaginatorium(a)despammed.com wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> As each ball n is removed, how many remain? >>>>>>> 9n. >>>>>>> >>>>>>>> Can any be removed and leave an empty vase? >>>>>>> Not sure what you are asking. >>>>>> If, for all n e N, n>0, the number of balls remaining after n's removal >>>>>> is 9n, does there exist any n e N which, after its removal, leaves 0? >>>>> I don't know what you mean by "after its removal"? >>>> Oh, I think this is clear, actually. Tony means: is there a ball (call >>>> it ball P) such that after the removal of ball P, zero balls remain. >>>> >>>> The answer is "No", obviously. If there were, it would be a >>>> contradiction (following the stated rules of the experiment for the >>>> moment) with the fact that ball P must have a pofnat p written on it, >>>> and the pofnat 10p (or similar) must be inserted at the moment ball P >>>> is removed. >> >>> I agree. If Tony means is there a ball P, removed at time t_P, such that >>> the number of balls at time t_P is zero, then the answer is no. After >>> all, I just agreed that the number of balls at the time when ball n is >>> removed is 9n, and this is not zero for any n. >> >>>> Now to you and me, this is all obvious, and no "problem" whatsoever, >>>> because if ball P existed it would have to be the "last natural >>>> number", and there is no last natural number. >>>> >>>> Tony has a strange problem with this, causing him to write mangled >>>> versions of Om mani padme hum, and protest that this is a "Greatest >>>> natural objection". For some reason he seems to accept that there is no >>>> greatest natural number, yet feels that appealing to this fact in an >>>> argument is somehow unfair. >> >>> The vase problem violates Tony's mental picture of a vase filling with >>> water. If we are steadily adding more water than is draining out, how >>> can all the water go poof at noon? Mental pictures are very useful, but >>> sometimes you have to modify your mental picture to match the >>> mathematics. Of course, when doing physics, we modify our mathematics to >>> match the experiment, but the vase problem originates in mathematics >>> land, so you should modify your mental picture to match the mathematics. >> >> As someone else has pointed out, the "balls" and "vase" >> are just an attempt to make this sound like a physical problem, >> which it clearly is not, because you cannot physically move >> an infinite number of balls in a finite time. It is just >> a distraction. As you say, the problem originates in mathematics. >> Any attempt to impose physical constraints on inherently unphysical >> problem is just silly. >> >> The problem could have been worded as follows: >> >> Let IN = { n | -1/(2^floor(n/10) < 0 } >> Let OUT = { n | -1/(2^n) } >> >> What is | IN - OUT | ? >> >> But that would not cause any fuss at all. >> >> Stephen >> > It would still be inductively provable in my system that IN=OUT*10. So you actually think that there exists an integer n such that -1/(2^floor(n/10)) < 0 but -1/(2^n) >= 0 ? What might that integer be? Stephen |