An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 26 Oct 2006 11:08 David Marcus 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. > I disagree. When you formulate a theory, whether scientific or mathematical, the goal should be to draw conclusions in line with observations. In science, it's no problem to disprove a theory, if there is a verifiable situation which it predicts incorrectly. When it comes to math, there is no such test, but the whole of mathematics should be consistent, and where one theory contradicts another, that's an indication that one or the other is less than correct. In the case of questions regarding oo, no theory should cause blatant contradictions, such as an event occurring but there being no moment in time during which it is occurring. If you have to accept such a conclusion to salvage a theory, it's time to look for alternatives that don't require you to sacrifice common sense and basic logic. This is just one example of where this theory goes wrong, along with proper subsets of the same size as the superset, and the concept of a smallest infinity. I simply don't accept the theory, because its conclusions are bizarre.
From: Tony Orlow on 26 Oct 2006 11:11 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.
From: Tony Orlow on 26 Oct 2006 11:12 Virgil wrote: > In article <453faeb8(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David R Tribble wrote: > >>> Tony Orlow wrote: >>>> Does anything occur in the vase at noon? If not, then it should have the >>>> same state as before noon. >>> As which state before noon? >>> >> The state of non-emptiness that persists continually from t>=-1 until t<0. > > What does TO mean by "from t >= -1"? > Does TO mean the same as "from t = -1"? > If so why not simply say so, and if not what does TO mean by it? > > > And even more puzzling, what does TO mean by "until t < 0"? > > Since "t < 0" is true before the experiment starts, TO must mean from > the beginning of time. I mean during the interval [-1,0). Any less puzzled? Probably not.
From: Tony Orlow on 26 Oct 2006 11:20 Virgil wrote: > In article <453fb285(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: >>> Randy Poe wrote: > >>>> 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. Events >> occurring in time must occupy at least one moment. > > > How is time irrelevant when every action is specified by the time at > which it is to occur? Please specify the moment when the vase becomes empty. > > The only relevant question is "According to the rules set up in the > problem, is each ball inserted at a time before noon also removed at a > time before noon?" > > An affirmative answer confirms that the vase is empty at noon. Not if noon is proscribed the the problem itself, which it is. > A negative answer directly violates the conditions of the problem. > > How does TO answer this question? > > As usual, he avoids such relevant questions in his dogged pursuit of the > irrelevant. > Noon does not exist in the experiment, or else you have infinitely numbered balls. >>> 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. The number of >> balls is related to the time by a formula which works in both directions. > > As time is a continuum and the numbers of balls in the vase is not, > there is no way of inverting the realtionship in the way that TO claims. Your times are as discontinuous as the number of balls, if no events can happen at any other moments than those specified. >> 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, and how? > > The vase is empty when every ball has been removed, and that occurs at > noon. So, that occurs AT noon? The vase becomes empty, when no balls are being removed? Remember, every ball was removed BEFORE noon, and upon the removal of each and every ball, more balls resided still in the vase. So, how does the vase empty, when no balls are removed? >>> If you follow the sequence of insertions and removals you never "get >>> to" noon but this doesn't imply that noon is never reached, or that >>> iterations involving non-naturally numbered balls occur. It just >>> implies that all insertion and removal is performed before noon. >>> >>> Tony won't let himself understand this. He is delusional. His problem. >>> >> I won't let myself accept self-contradictory conclusions. > > At least not unless they are TO's own personal self-contradictory > conclusions. Like the existence of balls in a vase from which all balls > have been removed. > > Like something occurring in time without at least a moment in which it occurred. > > There is no >> moment at which this can occur. The problem is perfectly modeled by a >> divergent infinite series. No last ball can be removed without there >> having been an negative number of balls previously. You solution fails. > > TO's assumption that there must be a last ball removed in order for all > balls to have been removed is part and parcel of his persistent delusion > that there must be a last (finite) natural number in order to have a set > of all (finite) natural numbers. No, that's what the problem implies when it claims to have completed the sequence of naturals. The fact is, with only naturally-numbered balls, one cannot have 1/n=0, and noon cannot be part of the experiment.
From: Tony Orlow on 26 Oct 2006 11:21
Virgil wrote: > In article <453fb2e8(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: >>>>>>>> stephen(a)nomail.com wrote: >>>>>>>>> Also, supposing for the sake of argument that there are "infinitely >>>>>>>>> number balls", if a ball is added at time -1/(2^floor(n/10)), and >>>>>>>>> removed >>>>>>>>> at time -1/(2^n)), then the balls added at time t=0, are those >>>>>>>>> where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 >>>>>>>>> then -1/(2^n) = 0 (making some reasonable assumptions about how >>>>>>>>> arithmetic >>>>>>>>> on these infinite numbers works), so those balls are also removed at >>>>>>>>> noon and >>>>>>>>> never spend any time in the vase. >>>>>>>> Yes, the insertion/removal schedule instantly becomes infinitely fast >>>>>>>> in >>>>>>>> a truly uncountable way. The only way to get a handle on it is to >>>>>>>> explicitly state the level of infinity the iterations are allowed to >>>>>>>> achieve at noon. When the iterations are restricted to finite values, >>>>>>>> noon is never reached, but approached as a limit. >>>>>>> Suppose we only do an insertion or removal at t = 1/n for n a natural >>>>>>> number. What do you mean by "noon is never reached"? >>>>>> 1/n>0 >>>>> Sorry, I meant t = -1/n. So, I assume your answer is that -1/n < 0. >>>>> >>>>> But, I don't follow. Translating "-1/n < 0" back into words, I get "all >>>>> insertions and removals are before noon". However, I asked you what >>>>> "noon is never reached" means. Are you saying that "noon is never >>>>> reached" means that "all insertions and removals are before noon"? >>>> Yes, David. What else happens in this experiment besides insertions and >>>> removals of naturals at finite times before noon? If the infinite >>>> sequence of events is actually allowed to continue until t=0, then you >>>> are talking about events not indexed with natural numbers, so you're not >>>> talking about the same experiment. If noon is not allowed, and all times >>>> in the experiment are finitely before noon, well, at none of those times >>>> does the vase empty, as we all agree. This is why I am asking when this >>>> occurs. It can't, given the constraints of the problem. >>> Does your "Yes" at the beginning of your reply mean that you agree that >>> "noon is never reached" means that "all insertions and removals are >>> before noon". By "mean", I mean that that is what the words mean, not >>> that the two statements are equivalent or deducible from each other. >>> >> Yes, every event, every insertion or removal, happens at a specific time >> before noon. At each of those times, the vase is non-empty. Nothing else >> occurs, as far as insertions of removals. Is that clear enough? So, when >> does the vase become empty, and how? > > The vase becomes empty in the usual way, by having everything in it > removed. And the time at which that finally has occurred is noon. Nothing is removed at noon. > > The only relevant question is "According to the rules set up by the > problem, is each ball inserted before noon also removed before noon?" > > An affirmative answer confirms that the vase is empty at noon. > A negative answer directly violates the conditions of the problem. > > How does TO answer? That you are a broken record, and noon does not exist in the experiment. It is never the case that every ball inserted has been removed. |