An uncountable countable set
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From: David Marcus on 2 Nov 2006 00:08 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. Sure. Consider the following translation of what you said into mathematics. 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). Then it is true (and easy to prove using freshman Calculus) that 1') g(x) > 0 for -1 <= x < 0. 2') For all j, b_j <> 0. 3') If t1 < t2, g(t1) > 0, and g(t2) = 0, then there is a j such that t1 < b_j <= t2. However, the 1'-3' don't imply 4') g(0) > 0. In fact, g(0) = 0 (as is easy to prove using freshman Calculus). So, 1'-3' don't imply 4'. Translating back to English, we conclude that your 1-3 don't imply that the vase is non-empty at noon. -- David Marcus
From: stephen on 2 Nov 2006 00:19 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
From: Virgil on 2 Nov 2006 00:23 In article <454973c5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Mike Kelly wrote: > >>> Tony Orlow wrote: > >>>> Mike Kelly wrote: > >>> <snip> > >>>>> 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. > >>> 1) It's not clear to me what you mean by that phrase but I'll assume > >>> the standard definition. Still, the question remains of which balls you > >>> think are in the vase? Does every natural number, n, have a ball in the > >>> vase labelled with that n? > >> Conceptually, sure. > > > > Yes or no? What is the set of balls in the vase at noon? Which balls > > are in the vase and which are not? > > > >>> 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. > > 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. > > > > ? > > > >> They both end up with countably many balls in the vase at noon. > > > > For now, I am going to try to restrict myself to discussing this new > > experiment, because I want to understand what "noon doesn't exist" is > > supposed to mean. And, again, your answer is ambiguous. I asked which > > balls are in the vase at noon, not the cardinality of the set of balls > > in the vase at noon. I then asked whether "noon exists", not whether > > anything "happens" at noon. Please try answering the questions people > > actually ask; it aids in communication. > > > >> The experiment's stated sequence logically precludes that the vase become > >> empty. > > > > 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. Wrong! The vase starts empty. > 2) No removals occur at noon. > 3) The vase can only become empty, after having contained balls, though > removal of balls. For the vase to become empty again, once balls have been inserted, it is only necessary that each ball inserted be removed. And before noon each one is.
From: Virgil on 2 Nov 2006 00:25 In article <454973f1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. Something that exists only in TO's dream world, is of no use anywhere else and of little use there except as a diversionary tactic. TO only brings it up when he his backed into corners.
From: Virgil on 2 Nov 2006 00:29
In article <45497460(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> 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. Except that TO cannot find any reason consistent with ZF or NBG or any standard set theory or any standard analysis, or even Robinson's non-standard analysis, for objecting, so he has to make up fairy tales like "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". It adds up nicely to no balls in the vase at and after noon. At least if one follows the rules of the gedankenexperiment. |