An uncountable countable set
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From: Virgil on 27 Oct 2006 00:01 In article <45417964(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Are you saying that V(0) is not equal to zero? > > > (sigh) I already answered this. Are you just trying to test for > consistency in my statements? That gets a little tiresome. It is equally tiresome trying to figure out why TO claims that after every ball inserted into a vase has been removed, TO should still claim that there are balls in that vase. > > I am saying 0 doesn't happen in this experiment. Then nothing else can happen either, since everything is tied to times and times are all tied to noon. Without noon, there is no experiment. > ... all finite balls would be gone, but they would be > replaced by an uncountable number of infinitely-numbered balls. Which spring out of TO's corrupted imagination. > > At the time of each and every event before 0, without exception, more > balls are left in the vase than were there before the event, and during > (-1,0), the vase is never empty. But it is at t = 0. > > Combine these two facts, and you get that the vase did not become empty, The only relevant question is "According to the rules set up in 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 overcome that inevitability? By claiming that noon never comes. But since all insertions/removals are tied to noon by their times, without noon none of them happen, and the vase is always empty.
From: Tony Orlow on 27 Oct 2006 00:02 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: >>>>> 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 >>> > >> How do you glean that from what I said? Your "largest finite" arguments >> are very boring. > > How do I glean that? You claim that IN does not equal OUT. > IN contains all n such that > -1/(2^floor(n/10)) < 0 > and OUT contains all n such that > -1/(2^n) < 0 > > Your claim that IN=OUT*10 (I am guessing you meant |IN|=|OUT|*10), > so presumably IN is bigger than OUT, and IN contains elements > that are not in OUT. The only way n can be an element of IN, > but not of out is if > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 Incorrect. For every n, finite or infinite, when the nth ball is removed, 9n remain. Either you get to t=0, in which case all finite balls are indeed gone, but replaced by uncountably many infinite balls, or you don't get to noon. > > So apparently you do not think such an n exists, yet you > think there are elements in IN that are not in OUT. > Those are contradictory positions. No, it is the only conclusion consistent with the notion that a proper subset is alway smaller than the superset. It is contradictory to the notion that a simple bijection indicates equal size for infinite sets. The mapping formulas must be taken into account for proper comparison in that case. > > Stephen Tony
From: Tony Orlow on 27 Oct 2006 00:05 Virgil wrote: > In article <4540cf58(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > That depends. > > If the apparently contradictory results follow from different axiom > systems, they may both be quite valid. > They cannot be mutually consistent, and a larger mathematical system including both cannot be internally consistent. The universe is consistent, and math creates and describes it. I'll take a side of logic, and hold the contradictions. > > >> 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. > > And we do not accept TO's theories for quite the same reason, his > conclusions are too bizarre. And many of his assumptions are too bizarre > as well. Sure. That's kind of relative. I'm used to that.
From: Tony Orlow on 27 Oct 2006 00:07 MoeBlee wrote: > Tony Orlow wrote: >> I share your and Godel's concerns about point set theory > > Oh how rich. How veddy veddy scholarly Mr. Orlow sounds when he says > such things, "I share Godel's concerns about point set theory." Too bad > Mr. Orlow doesn't know a single ding dang thing about Godel, or Godel's > concerns, or mathematical logic, or set theory, or point set topology, > or topology. > > MoeBlee > Wow, Lester's really getting under your skin, isn't he? He cracks me up. :)
From: Virgil on 27 Oct 2006 00:22
In article <45417cb7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > Ahem. I said that Robinson's analysis seems to have nothing to do with > transfinitology. They appear to be unrelated. However, they cme to two > very different conclusions regarding a basic question: is there a > smallest infinite number? Depends on what set of "numbers" one is allowing. In Robinson's set of non-standard reals, there is no least infinite. In ZF's ordinals, there is a least infinite. > It seems clear to me there is not, for the > very same reason that Robinson uses: if there is an infinite number, you > can subtract 1 and get a different, smaller infinite number. In Robinson's set of numbers one can do 0 - 1 and get a negative, but not in t ZF's ordinals, so what works in the one does not always work in the other. Thus TO's deliberate conflation of the two systems is deliberate fallacy. > So, that's a verrry basic discrepancy. There is clearly a contradiction > between the two theories. They can't both be right about that, can they? Yes. Robinson can be correct in his own set of "numbers" and ZF correct in its own set of "numbers" with no problems at all, as they are disjoint sets with distinct properties. > > Tach me more about mathematical logic and consistency. TO shows himself invincibly ignorant in such matters. > > In transfinitology? Why would anyone argue against me saying there is no > smallest infinite? Sorry, you're not meshing. We argue against deliberate misrepresentation. There are number systems that do not have any infinites. There are number systems that have a smallest infinite. There are number systems that have many infinites but no smallest one. Without a clear statement of what number system one is working in, one cannot validly make any statement about whether infinites can exist or whether there is a smallest one. > Geeze, calm down. I am not conflating them as if they were the same > thing. I am clearly stating that they are obviously mutually > incompatible, and you're too flustered by the disintegration of this > magical theory to see that simple point. They can't coexist in a > consistent universe. Diverse mutually exclusive number systems coexist in a consistent universe. They are merely disjoint as sets of numbers. But as TO's mind is too small for more that one system at a time to exist in his universe, he gets confused. I > > You're a damn fool, fooling only yourself, by thinking that you can > > shortcut the basics of the subject while you spout ignorantly on the > > Internet about the subject. > > > > MoeBlee > > > > Well, at least I'm not being such an obnoxious jerk. In that too TO is mistaken. > Or, maybe you think > I am. Read what Robinson says, and think about it. If you're really so > interested in "alternative" theories, I'd think you'd go to the source. I have read it, and it does not justify the wild claims TO makes. |