An uncountable countable set
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From: Tony Orlow on 8 Oct 2006 21:57 Virgil wrote: > In article <452949b7(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <45286ce5$1(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> David R Tribble wrote: >>>>> What about: >>>>> sum{n=0 to oo} (10n+1 + ... + 10n+10) - sum{n=1 to oo} (n) >>>>> The left half specifies the number of balls added to the vase, and >>>>> the right half specifies those that are removed. >>>>> >>>> Do you mean: >>>> sum{n=0 to oo} (10) - sum{n=0 to oo} (1)? >>>> That sounds like what you re describing, and termwise the difference is >>>> sum(n=0 to oo) (9). That's infinite, eh? >>> But the sums are not given termwise in the question, but sumwise, so >>> cannot be calculated termwise in your answer, but must be done sumwise. >>> >>> And sumwise they are no different. >> That's bass-ackwards. The gedanken specifically states that the >> insertion of 10 and the removal of 1 are coupled as an iteration in the >> process under discussion. > > As we are considering the expression suggested by David Tribble, > > <quote> > What about: > sum{n=0 to oo} (10n+1 + ... + 10n+10) - sum{n=1 to oo} (n) > The left half specifies the number of balls added to the vase, and > the right half specifies those that are removed. > <\quote> > > The original problem is, for the moment, irrelevant. > What David wrote didn't even make sense to me, which is why I re-expressed it. What does {10n+1+...+10n+10} mean except 100n+55, and what does that signify? Why is he summing this expression from 0, and summing n from 1? Are we really summing n, anyway? We're not adding the numbers on the balls to each other, we're just counting them 10- or 1 or 9-at-a-time. So, it's not the original problem which is irrelevant, but this meaningless expression which is irrelevant to the original problem. > >> Between every removal of 1 and the successive >> removal of 1 is an insertion of 10. So, each term '+10' must be coupled >> with a term '-1', which together make a '+9' per iteration. Where you >> violate the condition of the experiment that specifies this coupling of >> events into an iteration, you create your "discontinuity" at noon, and >> the monster under your bed. > > Let A_n(t) be equal to > 0 at all times, t, when the nth ball is out of the vase, > 1 at all times, t, when the nth ball is in the vase, and > undefined at all times, t, when the nth ball is in transition. > > Note that noon is not a time of transition for any ball, though it is a > cluster point of such times. Does time obey the law of trichotomy? Can something occur, not before, not after, and not at the same time as another event, given that the events happen instantaneously? > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > at any non-transition time t.Let A_n(t) be equal to > 0 at all times, t, when the nth ball is out of the vase, > 1 at all times, t, when the nth ball is in the vase, and > undefined at all times, t, when the nth ball is in transition. What is "in transition". Can't we consider the addition or removal of a ball to be instantaneous? > > Note that noon is not a time of transition for any ball, though it is a > cluster point of such times. > > let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > at any non-transition time t. > > B(t) is clearly defined and finite at every non-transition point, as > being, essentially, a finite sum at every such non-transition point. Correct, a linearly increasing finite value. > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > > > B(t) is clearly defined and finite at every non-transition point, as > being, essentially, a finite sum at every such non-transition point. > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 I understand your logic, but the basis is incorrect. As WM understandably complains, using the "completed set of naturals" as a measure of anything simply does not work. You're focused on the Twilight Zone.
From: Tony Orlow on 8 Oct 2006 22:07 Lester Zick wrote: > On Sun, 08 Oct 2006 15:12:28 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Sat, 07 Oct 2006 23:45:23 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>> Tony Orlow schrieb: >>>>> >>>>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>>>> Tony Orlow schrieb: >>>>>>> >>>>>>> >>>>>>>>>> Why not? Each and every number of the list terminates. That one is a number >>>>>>>>>> that does *not* terminate. >>>>>>>>>> >>>>>>>>>> > If you think that 0.111... is a number, but not in the list, >>>>>>>>> It is me who insists that it is not a representation of a number. >>>>>>>> Well, Wolfgang, that sets us apart, though I agree it's not a "specific" >>>>>>>> number. It's still some kind of quantitative expression, even if it's >>>>>>>> unbounded. Would you agree that ...333>...111, given a digital number >>>>>>>> system where 3>1? >>>>>>> That is the similar to 0.333... > 0.111.... But all these >>>>>>> representations exist only potentially, in my opinion. The difference >>>>>>> is, that 0.333... can be shown to lie between two existing numbers, so >>>>>>> we can calculate with it, while for ...333 this cannot be shown. >>>>>> I think it can be shown to lie between ...111 and ...555, given that >>>>>> each digit is greater than the corresponding digit in the first, and >>>>>> less than the corresponding digit in the second. >>>>> Yes, but only if we define, for instance, >>>>> >>>>> A n eps |N : 111...1 < 333...3 where n digits are symbolized in both >>>>> cases. >>>>> >>>>> This approach would be comparable with the "measure" which gives >>>>> >>>>> A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. >>>>> >>>>> I don't know whether these definitions are of any use, but I am sure >>>>> that they are not less useful than Cantor's cardinality. >>>>> >>>>> Regards, WM >>>>> >>>>> . >>>>> >>>> My opinion about that is, if one wants to talk about what happens "at >>>> infinity", that's the way that makes sense, not the measureless way of >>>> abstract set theory. I trust limit concepts, but not limit ordinals. >>> Tony, would it be fair to characterize what you're trying to say as >>> that there is some kind of positive/negative crossover at infinity >>> such that {-00, . . .,-1, 0, +1, . . . +00}? I haven't really been >>> following this thread too closely so I'm trying to understand what >>> you're after here in basic terms instead of the exact arguments >>> involved. >>> >>> ~v~~ >> Hi Lester, how's thangs? > > Hey, Tony, pretty much as usual. > >> I wasn't saying that right here, but agreeing with Wolfgang that limit >> concepts make sense, while the transfinitological approach doesn't. What >> I did say was that there are two ways to view the number line, one where >> oo and -oo are polar opposites, and a number circle where they are the >> same. > > But there is no single real number line. There's a single rational/ > irrational line but not even a single transcendental line, Tony. So on > the surface I don't see what this speculation has to recommend it. And > more to the point I don't see any way to effect a crossover in > mechanical terms. I've certainly heard you discuss your views on the number line, and how pi lies on a curve and rationals lie on straight lines, etc. To me, it sounds like a matter of construction, or meaning of the number, but not one of raw quantity. In terms of raw quantity on the real number line, they all obey the law of trichotomy, for any a and b, either a=b, a>b, or a<b. So, it's a linear order. > >> When it comes to reality, pretty much everything is in circles, >> including finite number systems so this model makes some sense, even if >> it doesn't in terms of limits of, say, powers. lim(x->-oo)=0 and >> lim(x->oo)=oo for n^x where n>1, so there the two are different. > > Well in a way I'm not sure I disagree. I'm interested in this aspect > of the problem mainly because I think that perhaps the issue both of > you may be trying to address is the closing of otherwise open sets. > Remember x->0 or plus or minus 00 doesn't mean x ever gets there. Perhaps not, but if, say, y=1/x is both oo and -oo at x=0, and oo=-oo, then the function is continuous in every respect, which is what we might desire in such a fundamental algebraic relation. > >> When it comes down to this argument, Wolfgang's argument, I agree with >> his logic concerning the naturals and the identity function between >> element count and value. > > To me "element count" and the number of commas are the same. Sure, that sounds okay to me. In the naturals, the first is 1, the second 2, etc. What is the aleph_0th? > >> He chooses then to reject infinities, while I >> choose to retool them. I think the problem he and I both see clearly is >> that using finiteness as a "bound" on the set simply doesn't tell you >> anything, but rather clouds the entire subject. >> >> Did that answer your question? > > Kinda, Tony, although I really don't get all the arguing over vases > and balls. And I don't really see finiteness as an issue if you view > infinity as the number of infinitesimals between limits. Then > infinities are contained whatever their magnitude. > > ~v~~ Yes, that's pretty much how I see them, the workable infinities, anyway. Pure infinity isn't really a number, any more than 0 is really a quantity. The ball and vase show comes from an example showing how ludicrous the standard take is on the question, given set theory. :) Tony
From: Tony Orlow on 8 Oct 2006 22:10 Virgil wrote: > In article <45296779(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <452946ad(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> Less than any finite distance. Silly! >>>>> And what is the smallest finite distance? >>>>> >>> Note question not answered!! But the correct answer of zero would have >>> blown TO's argument to blazes, so one can see why he would not care to >>> answer it. >> I was away a couple days, but I answered this, not paying attention to >> that apparent contradiction, since I don't consider 0 really a finite >> number at all. It's a point with no measure, as every number is measured >> relative to that point. > > If zero is not a number, how does TO keep the positive numbers separated > from negatives? It's not a finite number. It's the origin. A finite number is a finite distance from the origin. The origin is no distance from itself. >>>> When you claim that there are ordinals greater than any finite ordinal, >>>> are you obligated to name the largest finite ordinal? >>> When you claim there is a LUB to the reals strictly between 0 and 1, >>> are you required to name the largest real strictly between 0 and 1? >> No. That's my point. Why should I name the smallest object which is not >> infinitesimal? > > That is not at all what I asked. So TO is doing his STRAW MAN fallacy > thing again. You have no clue what the line of discussion was at this point, do you? >>> A "LUB" of the naturals does not have to be a natural any more than the >>> LUB of the the reals strictly between 0 and 1 has to be a real strictly >>> between 0 and 1. >> If it's a discrete set, then I disagree. > > The set {(n-1)/n: n in N} is a discrete set with a LUB which is not a > member of the set. In fact every strictly increasing sequence having a > LUB has a LUB which is not a member of the sequence. That is not "the reals strictly between 0 and 1" but a subset thereof.
From: Tony Orlow on 8 Oct 2006 22:25 Virgil wrote: > In article <45296f69(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David R Tribble wrote: >>>>> Virgil wrote: >>>>>>> Except for the first 10 balls, each insertion follow a removal and with >>>>>>> no exceptions each removal follows an insertion. >>>>> Tony Orlow wrote: >>>>>>> Which is why you have to have -9 balls at some point, so you can add 10, >>>>>>> remove 1, and have an empty vase. >>>>> David R Tribble wrote: >>>>>>> "At some point". Is that at the last moment before noon, when the >>>>>>> last 10 balls are added to the vase? >>>>>>> >>>>> Tony Orlow wrote: >>>>>> Yes, at the end of the previous iteration. If the vase is to become >>>>>> empty, it must be according to the rules of the gedanken. >>>>> The rules don't mention a last moment. >>>>> >>>> The conclusion you come to is that the vase empties. As balls are >>>> removed one at a time, that implies there is a last ball removed, does >>>> it not? >>> Please state the problem in English ("vase", "balls", "time", "remove") >>> and also state your translation of the problem into Mathematics (sets, >>> functions, numbers). >>> >> Given an unfillable vase and an infinite set of balls, we are to insert >> 10 balls in the vase, remove 1, and repeat indefinitely. In order to >> have a definite conclusion to this experiment in infinity, we will >> perform the first iteration at a minute before noon, the next at a half >> minute before noon, etc, so that iteration n (starting at 0) occurs at >> noon-1/2^n) minutes, and the infinite sequence is done at noon. The >> question is, what will we find in the vase at noon? With just this >> information, it is rather clear that this constitutes an infinite >> series, +(10-1)+(10-1)+(10-1)+..., or +9+9+9+..., which clearly diverges >> to some infinite number, of balls, that is. > > Not necessarily. If balls have, like electrons, no individual identity, > one can argue as TO does, but if they have individual identities, the > result at noon can depend on the order in which the balls are removed. >> However, the gedanken as described contains other information. Each ball >> is marked with a unique natural number, so that there is a ball for >> every natural. Given this identification scheme, we are given two >> possible sequences of inserting 10 and removing 1 which differ only in >> the labels on the balls. In both, we insert 1-10 on the first iteration, >> 2-20 on the second, etc, or balls 10n+1 through 10n+10 for each >> iteration (starting at 0). The difference is in the removal part of the >> iteration, where in the first case we remove 1, then 2, then 3, etc, or >> ball n+1 in iteration n. In the second case we remove ball 1, then 11, >> then 21, etc, or ball 10n+1 in iteration n. Because transfinite set >> theory makes no distinction between aleph_0 and aleph_0/10, no >> distinction is made between the number of iterations it takes to add all >> the balls, and how many it takes to remove them. Erroneously, the >> unending sequences of required iterations are considered the same. So, >> in the first case, it is reasoned that all balls are inserted before >> "noon" and all balls are removed before "noon", and therefore nothing >> remains at noon. In the second case, balls 2 through 10 clearly remain, >> and all n+2 through n+10, for all n in N, which is an infinite set of >> balls remaining. So, apparently, one might surmise that you could >> perform experiment 2, have some infinite number of balls in the vase, >> and then switch the labels around afterwards and make them all >> disappear. It's just silliness. > > > TO's misrepresentation is the "mere silliness" >> If the answer is clear with limited information, then additional >> information is irrelevant. > > But the answer depends on two things, the individuality of the balls, > and given that they are individual, the order of removal. > > If TO assumes that they have no individuality, as he does, then giving > them individuality changes the problem. And that different problem may > have a different solution. > >> With them, well, we have a bunch of hocus >> pocus, and I can't even really give any mathematical expression to that. > > > I can: > > Let A_n(t) be equal to > 0 at all times, t, when the nth ball is not in the vase, > 1 at all times, t, when the nth ball is in the vase, and > undefined at all times, t, when the nth ball is in transition. > > Note that noon is not a time of transition for any ball, though it is a > cluster point (limit point) of such times. > > Let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase > at any non-transition time t. > > B(t) is clearly defined and finite at every non-transition point, as > being, essentially, a finite sum at every such non-transition point of > the finitely many non-zero values. > > Further, A_n(noon) = 0 for every n, so B(noon) = 0. > Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > > > > > > >> It was easy without the labels. "Pay no attention to the man behind the >> vase. I am the Great and Powerful Aleph" ;) > > And it is a different problem without the labels. If noon is not a point of transition, then nothing can change at noon, and the vase is non-empty at every point before noon, so it must be non-empty at noon as well.
From: Dik T. Winter on 8 Oct 2006 22:49
In article <1160310273.517860.47380(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > Ok, but as we have agreement now, we can return to he main question: > > > Why do you think that 0.111... with the index sequences 1,2,3,... or > > > k+1,k+2,k+3 or -k, -k+1, -k+2, ... represents exactly *one* number > > > only, as you asserted? > > > > Why do you still maintain that I think it represents a number? How many > > times do I need to state that, without proper definition, it only is > > a sequence of symbols that I on occasion call a "number". Because I have > > not yet seen a definition of "number", and you have stated that you are > > not able to give one... > > In the decimal system of current mathematics 0.111... = 1/9 and is a > number without doubt. Yes, and in the octal system it is 1/7, and in the reversed 10-adics it is -1/9. There are a host of systems where it has a meaning, but it has no meaning if you do not specify the system. And there are also systems where it has no meaning because there is no convergence (the 10-adics in normal notation). > If omega does exist, then 0.111... has omega 1's. Yes. > But whatever. As a sequence of symbols, > > igoring the "0.", it is in bijection with N. It also is in bijection > > with {k+1,k+2,...} for every k. As {k+1,k+2,...} is in bijection with N. > > A bijection with N does not define the indexes such that there was only > a unique sequence. Therefore, there is not, as you asserted, one unique > number 0.111... . Oh. Whatever. Care to explain? (And, pray, dispense of the word number in this context.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |