An uncountable countable set
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From: Tony Orlow on 27 Oct 2006 11:34 imaginatorium(a)despammed.com wrote: > Virgil wrote: >> In article <45417528$1(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: > > <snip> > >>> For what it's worth, and I know this doesn't add a lot of credibility to >>> Ross in your eyes, coming from me, but I think Ross has a genuine >>> intuition that isn't far off with respect to what's controversial in >>> modern math. Sure, he gets repetitive and I don't agree with everything >>> he says, but his cryptic "Well order the reals", which I actually >>> haven't seen too much of lately, is a direct reference to his EF >>> (Equivalence Function, yes?) between the naturals and the reals in >>> [0,1). The reals viewed as discrete infinitesimals map to the >>> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >>> answer your question, I think Ross makes some sense. But, of course, >>> coming from me, that probably doesn't mean much. :) >> Coming from TO it damns Ross. > > Even by your standards, Virgil, this is egregiously silly. TO skips the > basic exposition in Robinson's book, but finds a sentence he likes. So > this "damns" Robinson's non-standard analysis, does it? > > Brian Chandler > http://imaginatorium.org > No, he's saying my endorsement of Ross Finlayson's ideas reduces his credibility, or some such. I didn't skip the "basic exposition". I skipped ahead from where he was talking about "stratified sentences", leaving a bookmark there to return, because it was very heady and technical. Have you read it? You should. Then talk. I have not found one sentence I like, but conclusion after conclusion all in line with what I've been saying, though I questioned one of his conclusions, without disagreeing. In all, he makes perfect sense so far. In any case, if you're going to respond to Virgil's silliest comments (I didn't bother with this one), you might want to pay attention to what he's being silly about. TOny
From: Tony Orlow on 27 Oct 2006 11:38 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: >>>>>>> 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. > > What are you talking about? I defined two sets. There are no > balls or vases. There are simply the two sets > > IN = { n | -1/(2^floor(n/10)) < 0 } > OUT = { n | -1/(2^n) < 0 } > For each n e N, IN(n)=10*OUT(n). >>> 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. > > But OUT is not a proper subset of IN, unless you believe that > there exists an n such that > -1/(2^floor(n/10)) < 0 > but > -1/(2^n) >= 0 > > If you claim that OUT is a proper subset of IN, you must > be able to identify an element that is in IN that is not in OUT. > > Stephen For each n e N, IN(n)=10*OUT(n). You appear to think that the times are all that matters, but the times included in IN each apply to 10 elements, whereas the times in OUT each apply to only one.
From: Ross A. Finlayson on 27 Oct 2006 11:38 Tony Orlow wrote: > The Inverse Function Rule uses infinite-case induction to finely order > infinite sets of reals mapped from a standard set, N. Where there is a > bijection between N and a set S using f(n)=s, there is a mapping from S > to N using g(s)=n, where g(f(x))=f(g(x)) (inverse functions for the > bijection). The size of the set S over the interval [a,b] is given by > floor(g(b)-g(a)+1). (I think I wrote that correctly). This works for all > finite sets of reals. The number of square roots, for instance, between > 1 and 100 is floor(100^2-1^2+1), 10000 square roots, from sqrt(1) to > sqrt(10000). IFR can easily be used to show that the evens are half as > numerous as the naturals, and other interesting "facts". > > EF is the special case of IFR mapping the naturals in [0,oo) to the > reals in [0,1), using the mapping function f(n)=n/oo. Isn't that how you > define the equivalency function? Given this mapping, we can say > g(s)=s*oo, so that over the entire real line, we have oo^2 reals, oo in > each unit interval, over oo unit intervals. Does that sound about right? > > Tony Isn't there symmetry about the origin thus it's 2 times oo^2? Obviously half of the integers are even. What are cases against use or validity of IFR? How do you address those? Ross
From: Tony Orlow on 27 Oct 2006 11:47 RLG wrote: > "Tony Orlow" <tony(a)lightlink.com> wrote in message > news:4540f9d0(a)news2.lightlink.com... >> This is very simple. Everything that occurs is either an addition of ten >> balls or a removal of 1, and occurs a finite amount of time before noon. >> At the time of each event, balls remain. At noon, no balls are inserted or >> removed. The vase can only become empty through the removal of balls, so >> if no balls are removed, the vase cannot become empty at noon. It was not >> empty before noon, therefore it is not empty at noon. Nothing can happen >> at noon, since that would involve a ball n such that 1/n=0. > > > Tony, I think your confusion results from imagining the balls without any > labels. In this case at 1 minute before noon 10 balls are inserted into the > vase, at 1/2 minute before noon 9 balls are inserted into the vase, at 1/4 > minute before noon, 9 more balls are inserted into the vase and, in general, > at (1/2)^n minutes before noon 9 balls are inserted into the vase. So you > are saying that the number of vase balls at noon is: > > > 10 + 9 + 9 + 9 + 9 + 9 + ... = Infinite. > Yes, but I don't consider that confusion. If the problem is solvable without the labels, then the labels don't matter. > > Or, since one ball is removed each time ten more are added, we should write: > > > 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = Infinite. > > > Now, this divergent series is conditionally convergent. That means we can > make the sum equal any value we like depending on how the terms are > arranged. So if we choose 0 for the sum that is perfectly valid: > > > 10 + (10-1) + (10-1) + (10-1) + (10-1) + ... = 0. > No, we went through this in another thread. The only way to get a sum of 0 is by rearranging the terms and grouping so you have ten -1's for every +10. But, the sequence of events is specified NOT to be in that order. No ball can be removed without having ten inserted immediately before. So, despite the silly games that mathematicians may play with "conditionally convergent" series, none of that applies to the ball and vase problem as stated. Does that sound confused to you? > > In this case there are no balls in the vase at noon. Without labels on the > balls there is no criterion by which to select what the sum should be and > the end state of the supertask is undefined. As I noted in an earlier post, > if some of the balls are labeled with numbers that are not naturals, for > example transfinite ordinal numbers, we can choose "Infinite" for the sum if > the circumstances require it. > No, the order cannot be rearranged. For each iteration you have a net addition of nine balls. You cannot remove a ball without adding ten more. This is clearly the divergent sum(n=1->oo: 9). > > Consider the following problem: > > > Tony has a two gallon bucket and his job is to ensure that the amount of > water in the bucket during the nth day is 1+sin(n) gallons. Since Tony's > job never ends he will always be making daily changes in the bucket's water > content and we have a full mathematical description of Tony's job. There is > no problem with this. But if we changed Tony's job so that it had an end, > say at noon, and the bucket had to contain 1+sin(n) gallons at (1/2)^n > minutes before noon then we do not have a full description of Tony's > activities. It is a mistake to assume the bucket's water content at noon is > a function of its pre-noon state. At noon Tony puts whatever amount of > water he wants into the bucket. > > > -R > > > Now you are talking about a series which diverges due to oscillation, more or less. At noon the bucket would have to be filling and emptying infinitely quickly. So what? Clearly, at noon, it has somewhere between 0 and 2 gallons of water, but no specific quantity. That's a different problem.
From: Tony Orlow on 27 Oct 2006 11:48
imaginatorium(a)despammed.com wrote: > David Marcus wrote: >> imaginatorium(a)despammed.com wrote: >>> Virgil wrote: >>>> In article <45417528$1(a)news2.lightlink.com>, >>>> Tony Orlow <tony(a)lightlink.com> wrote: >>> <snip> >>> >>>>> For what it's worth, and I know this doesn't add a lot of credibility to >>>>> Ross in your eyes, coming from me, but I think Ross has a genuine >>>>> intuition that isn't far off with respect to what's controversial in >>>>> modern math. Sure, he gets repetitive and I don't agree with everything >>>>> he says, but his cryptic "Well order the reals", which I actually >>>>> haven't seen too much of lately, is a direct reference to his EF >>>>> (Equivalence Function, yes?) between the naturals and the reals in >>>>> [0,1). The reals viewed as discrete infinitesimals map to the >>>>> hypernaturals, anyway, and his EF is a special case of my IFR. So, to >>>>> answer your question, I think Ross makes some sense. But, of course, >>>>> coming from me, that probably doesn't mean much. :) >>>> Coming from TO it damns Ross. >>> Even by your standards, Virgil, this is egregiously silly. TO skips the >>> basic exposition in Robinson's book, but finds a sentence he likes. So >>> this "damns" Robinson's non-standard analysis, does it? >> Virgil said "Ross", not "Robinson", I believe. > > Yes, of course. But Virgil's implication is that "TO says person P is > right about something" implies P is wrong. This may, contingently, be > true about Ross, but the argument could equally be applied to Robinson, > in which case the conclusion is obviously not true. > > Brian Chandler > http://imaginatorium.org > And, what about those rare occasions when I agree with Virgil? Uh oh. |