An uncountable countable set
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From: Virgil on 3 Oct 2006 03:15 In article <4522007f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >>> For the sake of this argument, we can talk about infinite reals, of > >>> which infinite whole numbers are a subset. > > > > David R Tribble wrote: > >>> What are these "infinite reals" and "infinite whole numbers" that you > >>> speak of so much? > > > > Tony Orlow wrote: > >>> The very same, with no restriction of finiteness. Any T-riffic number > >>> has successor. :) > > > > David R Tribble wrote: > >>> It's sporting of you to drop the requirement that all the naturals in N > >>> have to be finite, but since all of them are, it's meaningless to say > >>> "with no restriction of finiteness". That's kind of like saying N > >>> contains all naturals "with no restriction of non-integer values". > >>> I can say that, but it does not change the fact that all the members > >>> of N are integers. > > > > Tony Orlow wrote: > >> Is the successor to ...11110000 not equal to ...11110001? > > > > I don't know - how are you defining those numbers? > > > > Even if it is (or is not), what does it have to do with the finite > > naturals? Are you saying that those "infinite naturals" are > > somehow successors to the finite naturals? How? > > > > > > David R Tribble wrote: > >>> So I ask again, where are those infinite naturals and reals you keep > >>> talking about? It's obvious they are not in N. > > > > Tony Orlow wrote: > >> [No] it's not. > > > > Every member of N has a finite successor. Can you prove that your > > "infinite naturals" are members of N? > > > > Yes, if "finite successor" is the only criterion. But it is not. In addition, N must be minimal with respect to closure under successorship, which excludes TO's "infinite naturals". > > To prove finiteness of such a string: > > The bits over each sequence are indexed by natural numbers, which are > all finite, yes? > > For any finite bit position, the string up to and including that bit > position can only represent a finite value, yes? > > Therefore, there is no bit position where the string can have > represented anything but a finite value, see? If the length is > potentially, but not actually, infinite, so with the value. That is true of the members, but not of the set itself. > > > To prove successorship of such a string: > > The rule for successorship for finite values is > 1. Find the rightmost (least significant) 0 > 2. Invert from that 0 rightwards > > This works for all values where there is a rightmost 0. That excludes > ...111, which can only have successor given ignored overflow, allowable > in some cases. But not in N. > > You don't really question why the successor to ...11110000 is equal to > ...11110001, do you? We question why TO thinks that has anything to do with natural numbers. Von Neumann's N is such that (1) {} is a member of N (2) If x is a member of N then so is x \union {x} (3) If S is an subset of N such that (3.1) {} is a member of S, and (3.2) if x is a member of S then so is x \union {x} then S = N. And that is, by general consent, a satisfactory model for N. But that model cannot contain any of TO's allegedly infinite naturals. (1) there cannot be a first infinite natural, and, (2) since this N can be proved to be well ordered, Therefore, every subset of N with no first element must be empty.
From: Han de Bruijn on 3 Oct 2006 03:18 Virgil wrote: > History is rife with mathematical developments that, at the time of > their development, had no uses in physics or any other science, but > which later turned out to be essential to some science's further > development. This is both true and misleading. Han de Bruijn
From: Virgil on 3 Oct 2006 03:22 In article <452202bd(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >>> You drew that from my suggestion of the number circle, and that ...11111 > >>> could be considered equal to -1. Since then, I looked it up. I'm not the > >>> first to think that. It's one of two perspectives on the number line. > >>> It's either really straight, or circular with infinite radius, making it > >>> infinitesimally straight. The latter describes the finite universe, and > >>> the former, the limit. But, you knew that, and are just trying to have > >>> fun. > > > > David R Tribble wrote: > >>> You are drawing geometric conclusions that are not warranted. > >>> The Projective Real Line is simply R U {oo}. Adding unsigned oo > >>> to the set allows certain arithmetic operations to be performed > >>> that are undefined in the regular real set. > >>> > >>> But simply adding the limit point oo to the set does not actually make > >>> it a "circle", because oo has no predecessor or successor, and > >>> certain operations like oo+1 and oo+oo are still meaningless within > >>> the set. > > > > Tony Orlow wrote: > >> You do realize that my statements involve a considerable amount of > >> personal reflection, don't you? There is more to the number circle than > >> "proven". In the binary number circle, "100...000" is both positive and > >> negative infinity. > > > > By what axioms? > > > > By the theorem that the negative of a 2's complement number is > calculated as the bitwise inverse plus 1. That may hold for finite strings, as in computer registers, but I see no evidence that it holds for endless strings anywhere. > > It's pretty obvious that the mathematically inclined among us follow > > what you're saying, because, honestly, it's really not that complicated > > or profound. Most of it also happens to be self-contradictory, which > > is where most of us have a problem "seeing" it. Which makes it > > incumbent upon you to define your ideas a little more clearly and > > a lot more formally. > > If you see a contradiction, please say what it is. I'm happy to discuss > whatever seems wrong. When we do explain, TO still does not deign to see. > > And if it's that basic (it's not meant to be complicated or profound) > then why isn't it mainstream? Self-contradictory things like TO's theories tend not to get promoted to mainstream. > > > > > It's not a failing on my part that your ideas are not consistent, but > > rather a failing on your part not to recognize the logical conclusions > > that prove these inconsistencies. > > > > What inconsistencies? The very ones ones you refuse to admit to however often they are pointed out to you by however many people.
From: Han de Bruijn on 3 Oct 2006 03:26 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>stephen(a)nomail.com wrote: > >>>Han.deBruijn(a)dto.tudelft.nl wrote: >>> >>>>Worse. I have fundamentally changed the mathematics. Such that it shall >>>>no longer claim to have the "right" answer to an ill posed question. >>> >>>Changed the mathematics? What does that mean? >>> >>>The mathematics used in the balls and vase problem >>>is trivial. Each ball is put into the vase at a specific >>>time before noon, and each ball is removed from the vase at >>>a specific time before noon. Pick any arbitrary ball, >>>and we know exactly when it was added, and exactly when it >>>was removed, and every ball is removed. >>> >>>Consider this rephrasing of the question: >>> >>> you have a set of n balls labelled 0...n-1. >>> >>> ball #m is added to the vase at time 1/2^(m/10) minutes >>> before noon. >>> >>> ball #m is removed from the vase at time 1/2^m minutes >>> before noon. >>> >>> how many balls are in the vase at noon? >>> >>>What does your "mathematics" say the answer to this >>>question is, in the "limit" as n approaches infinity? > > >>My mathematics says that it is an ill-posed question. And it doesn't >>give an answer to ill-posed questions. > > That is a perfectly reasonable answer. But you do agree that > for this problem, the vase is empty at noon for any finite n. > So one wonders what criteria you used to determine that > this infinity cannot be approached via limits. We can say that the number of balls Bk at step k = 1,2,3,4, ... is: Bk = 9 + 9.ln(-1/tk)/ln(2) where tk = - 1/2^(k-1) for all k in N . And that's ALL we can say. The version of the problem used here is the first experiment in: http://groups.google.nl/group/sci.math/msg/d2573fcb63cbf1f0?hl=en& Han de Bruijn
From: Virgil on 3 Oct 2006 03:27
In article <452203a9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452160ef(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Randy Poe wrote: > > > >>> You're right. So it's back to Tony. The quote is yours (Tony), so > >>> just follow the attributes back to "if noon exists, that's > >>> when the vase empties", and that's when you (Tony) > >>> said such a thing. > >>> > >>> - Randy > >>> > >> Okay Randy, but does it, or not? Is it non-empty, then empty? Does this > >> change in state occur before noon? After noon? Or, at noon? > > > > It is at a particular state at any particular time, but as the states do > > not form a continuum, one gets jump discontinuities in the number of > > balls in the vase in a neighborhood of each relevant time. And in every > > neighborhood of noon, there are infinitely many such discontinuities. > > Yo! In da House! Diss! Diss! Diss! Discontinuities! > > This is the problem with the ordinals. They're not "real". There is no > "line". The problem with TO is that nothing he does is real. > > Time to let von Neumann's snow job slide. Von Neumann's work is a good deal more widely accepted and acceptable than TO's multiple snow jobs. TO should look up von Neumann's bio and find out how much besides a model for N he created. Then go off some place and sulk for a while, cause TO ain't ever going to come close to even understanding all vN created. Global warming and all. Toodles. > > Tony |