An uncountable countable set
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From: Tony Orlow on 1 Nov 2006 23:28 Mike Kelly wrote: > Tony Orlow wrote: >> stephen(a)nomail.com wrote: >>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> stephen(a)nomail.com wrote: >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>> stephen(a)nomail.com wrote: >>>>>>> <snip> >>>>>>> >>>>>>>>> What does that have to do with the sets IN and OUT? IN and OUT are >>>>>>>>> the same set. You claimed I was losing the "formulaic relationship" >>>>>>>>> between the sets. So I still do not know what you meant by that >>>>>>>>> statement. Once again >>>>>>>>> IN = { n | -1/(2^(floor(n/10))) < 0 } >>>>>>>>> OUT = { n | -1/(2^n) < 0 } >>>>>>>>> >>>>>>>> I mean the formula relating the number In to the number OUT for any n. >>>>>>>> That is given by out(in) = in/10. >>>>>>> What number IN? There is one set named IN, and one set named OUT. >>>>>>> There is no number IN. I have no idea what you think out(in) is >>>>>>> supposed to be. OUT and IN are sets, not functions. >>>>>>> >>>>>> OH. So, sets don't have sizes which are numbers, at least at particular >>>>>> moments. I see.... >>>>> If that is what you meant, then you should have said that. >>>>> And technically speaking, sets do not have sizes which are numbers, >>>>> unless by "size" you mean cardinality, and by "number" you include >>>>> transfinite cardinals. >>>> So, cardinality is the only definition of set size which you will >>>> consider.....your loss. >>> If somebody presents another definition of set size, I will >>> consider it. You have not presented such a definition. >>> >>> >> I have presented an approach that works for the majority of infinite >> bijections, and explained some of the exceptions. IFR works for all >> numeric sets mapped from a common set. N=S^L works for all languages, >> including those that express the first set. Both work on a parameteric >> basis, using infinite case induction to finely order the values of >> formulas for a specific infinite n. Rare exceptions include the set 1/n >> for neN, whose inverse is itself, which IFR ends up saying has size 1, >> but that's because the natural indexes and fractional mapped reals only >> share one point in their range, 1. So, I think Bigulosity is worth >> considering. > > Why? What is it good for? What theories is it used in? > Bigulosity Theory.
From: Tony Orlow on 1 Nov 2006 23:30 Mike Kelly wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David Marcus wrote: >>>>>>>>> Tony Orlow wrote: >>>>>>>>>> David Marcus wrote: >>>>>>>>>>> Tony Orlow wrote: >>>>>>>>>>>> David Marcus wrote: >>>>>>>>>>>>> Tony Orlow wrote: >>>>>>>>>>>>>> Mike Kelly wrote: >>>>>>>>>>>>>>> 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 admit that you might be wrong? >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> Let's start simply so there is less room for mutual incomprehension. >>>>>>>>>>>>>>> Let's imagine a new experiment. In this experiment, we have the same >>>>>>>>>>>>>>> infinite vase and the same infinite set of balls with natural numbers >>>>>>>>>>>>>>> on them. Let's call the time one minute to noon -1 and noon 0. Note >>>>>>>>>>>>>>> that time is a real-valued variable that can have any real value. At >>>>>>>>>>>>>>> time -1/n we insert ball n into the vase. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> My question : what do you think is in the vase at noon? >>>>>>>>>>>>>> A countable infinity of balls. >>>>>>>>>>>>> So, "noon exists" in this case, even though nothing happens at noon. >>>>>>>>>>>> Not really, but there is a big difference between this and the original >>>>>>>>>>>> experiment. If noon did exist here as the time of any event (insertion), >>>>>>>>>>>> then you would have an UNcountably infinite set of balls. Presumably, >>>>>>>>>>>> given only naturals, such that nothing is inserted at noon, by noon all >>>>>>>>>>>> naturals have been inserted, for the countable infinity. Then insertions >>>>>>>>>>>> stop, and the vase has what it has. The issue with the original problem >>>>>>>>>>>> is that, if it empties, it has to have done it before noon, because >>>>>>>>>>>> nothing happens at noon. You conclude there is a change of state when >>>>>>>>>>>> nothing happens. I conclude there is not. >>>>>>>>>>> So, noon doesn't exist in this case either? >>>>>>>>>> Nothing happens at noon, and as long as there is no claim that anything >>>>>>>>>> happens at noon, then there is no problem. Before noon there was an >>>>>>>>>> unboundedly large but finite number of balls. At noon, it is the same. >>>>>>>>> So, noon does exist in this case? >>>>>>>> Since the existence of noon does not require any further events, it's a >>>>>>>> moot point. As I think about it, no, noon does not exist in this problem >>>>>>>> either, as the time of any event, since nothing is removed at noon. It >>>>>>>> is also not required for any conclusion, except perhaps that there are >>>>>>>> uncountably many balls, rather than only countably many. But, there are >>>>>>>> only countably many balls, so, no, noon is not part of the problem here. >>>>>>>> As we approach noon, the limit is 0. We don't reach noon. >>>>>>> To recap, we add ball n at time -1/n. We don't remove any balls. With >>>>>>> this setup, you conclude that noon does not exist. Is this correct? >>>>>> I conclude that nothing occurs at noon in the vase, and there are >>>>>> countably, that is, potentially but not actually, infinitely many balls >>>>>> in the vase. No n in N completes N. >>>>> Sorry, but I'm not sure what you are saying. Are you saying that what I >>>>> wrote is correct or are you saying it is not correct? I'll repeat the >>>>> question: >>>>> >>>>> We add ball n at time -1/n. We don't remove any balls. With >>>>> this setup, you conclude that noon does not exist. Is this correct? >>>>> Please answer "yes" or "no". >>>>> >>>> What do YOU mean by "exist"? Does anything happen which is proscribed if >>>> noon DOES arrive? No, not in this case. So, noon case "exist" or not. >>> There is no event at noon. There is no "noon case". But you >>> seem to be saying that arrival of the actual time of noon, everywhere >>> in the world, is somehow controlled by how we define a certain set >>> of events. >>> >>> If you mean is there an event at noon, then say so. Don't say >>> "noon doesn't happen". >>> >>> There's an event at -60 seconds. The next event is at -30 seconds. >>> There's no event at -50 seconds. But would you really say >>> "-50 is proscribed in this experiment" or "-50 doesn't exist"? >>> >>>> In >>>> the other case, the vase also does not empty before noon, and nothing >>>> happens at noon. So, then, why do you conjecture that it's empty AT noon? >>> In the absence of any events happening at noon, we need to >>> define what is meant by "number of balls in the vase at noon". >>> >>> We define that as "number of balls which have been inserted >>> at t<=noon and not removed". >>> >>> Forget calling this the "number of balls in the vase at noon". That >>> bothers you. Will you allow us to discuss "the set of balls which >>> have been inserted but not removed?" >>> >>> - Randy >>> >> 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
From: Tony Orlow on 1 Nov 2006 23:39 Mike Kelly wrote: > Tony Orlow wrote: >> stephen(a)nomail.com wrote: >>> So if instead, someone had just posed this problem >>> Let >>> IN = { n | -1/2^(floor(n/10)) < 0 } >>> OUT = { n | -1/2^n < 0 } >>> >>> What is | IN - OUT | there would be controversy. >>> Note, there are not balls or vases, or times or anything >>> in this problem. Just two sets. It would help if you >>> bother to stop and think for a second before responding. >>> >> Where are the iterations mentioned there? You're missing the crucial >> part of the experiment. By your logic, you could put them in in any >> order and remove them in any order, and when you say both processes are >> done, nothing's left, but that's BS. It ignores the sequence specified. >> This is just a distraction. > > Yes, if you insert and remove exactly the same balls then you get the > same result when you're done, no matter what order you did it all in. > Why is that BS? It seems blindingly obvious. It's BS applied to the problem at hand, because sets entirely ignore the correspondence between insertions and removals over t or n, and try to use omega or aleph_0 as some actual number, when the Twilight Zone between the finite and the infinite, the largest finite or smallest infinite, is exactly like the Twilight Zone between the smallest positive real and 0. There is no smallest possible infinite number, or it would have a finite number finitely before it. You are burying the correspondence between t, n, and f(n) in the time vortex. It's a hat trick. Next.... > > But I forgot, you think that if you shift all the insertions 1 minute > further back in time, you DO get an empty vase at noon, right? I really > don't understand how your mind works. > You don't understand the contingency between a *removal of 1 and a subsequent *addition of 10 before any further *removals of 1? Where is that covered in your set-theoretic schema? I see no t in there, and only vague reference to n.
From: Tony Orlow on 1 Nov 2006 23:48 Randy Poe 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." > > I've asked questions similar to yours. The answer is: "It's easy > for me to figure out there's a ball in the vase at noon. Therefore > I will allow noon to 'exist' in this problem." > > - Randy > I've pointed out a couple contradictions in your position. 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. 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. If it doesn't happen at time t, or at any time s<t, then it hasn't happened BY time t. If the vase were to empty, by the removal of the maximum of one ball per t or n, that removal would have to have been (immediately in the n sense, or immediately in the t sense) preceded by the addition of ten balls, which means there would have had to have been -9 balls in the vase at the end of the last iteration, or event. Tony Orlow
From: David Marcus on 1 Nov 2006 23:56
Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> stephen(a)nomail.com wrote: > >>> So if instead, someone had just posed this problem > >>> Let > >>> IN = { n | -1/2^(floor(n/10)) < 0 } > >>> OUT = { n | -1/2^n < 0 } > >>> > >>> What is | IN - OUT | there would be controversy. > >>> Note, there are not balls or vases, or times or anything > >>> in this problem. Just two sets. It would help if you > >>> bother to stop and think for a second before responding. > >>> > >> Where are the iterations mentioned there? You're missing the crucial > >> part of the experiment. By your logic, you could put them in in any > >> order and remove them in any order, and when you say both processes are > >> done, nothing's left, but that's BS. It ignores the sequence specified. > >> This is just a distraction. > > > > Yes, if you insert and remove exactly the same balls then you get the > > same result when you're done, no matter what order you did it all in. > > Why is that BS? It seems blindingly obvious. > > It's BS applied to the problem at hand, because sets entirely ignore the > correspondence between insertions and removals over t or n, and try to > use omega or aleph_0 as some actual number, when the Twilight Zone > between the finite and the infinite, the largest finite or smallest > infinite, is exactly like the Twilight Zone between the smallest > positive real and 0. There is no smallest possible infinite number, or > it would have a finite number finitely before it. You are burying the > correspondence between t, n, and f(n) in the time vortex. It's a hat > trick. Next.... > > > But I forgot, you think that if you shift all the insertions 1 minute > > further back in time, you DO get an empty vase at noon, right? I really > > don't understand how your mind works. > > You don't understand the contingency between a *removal of 1 and a > subsequent *addition of 10 before any further *removals of 1? Where is > that covered in your set-theoretic schema? I see no t in there, and only > vague reference to n. As many people have pointed out, it is quite straightforward to include the sequence of additions and removals and the times at which they occur in the mathematics (presumably Littlewood started with some math like the following when he made up the problem): For j = 1,2,..., let a_j = -1/floor((j+9)/10), b_j = -1/j. For j = 1,2,..., define a function f_j: R -> R by f_j(x) = 1 if a_j <= x < b_j, 0 if x < a_j or x >= b_j. Let g(x) = sum_j f_j(x). What is g(0)? Since (according to freshman Calculus), g(0) = 0, we (once again) conclude that the vase is empty. In fact, I don't see any way of translating the problem into mathematics which gives any answer other than zero. Of course, if you wish to start with the assumption that the vase is not empty, you can conclude anything you want. -- David Marcus |