An uncountable countable set
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From: Tony Orlow on 3 Oct 2006 02:48 David R Tribble wrote: > Virgil wrote: >>> According to TO every set of numbers has a natural order, and it is >>> within that natural order that we must view it, but now he wants to >>> reject the natural order because it runs counter to another of his >>> claims. > > Tony Orlow wrote: >> I'm saying the if you iterate the negative integers starting at 0, in >> that order, there is no infinite descending sequence. On the other hand >> I don't know why I said "neither can the reals". In any case, the only >> way the ordinals manage to be "well ordered" is because they're defined >> with predecessor discontinuities at the limit ordinals, including 0. >> That doesn't seem "real", and the axiom of choice aside, I don't see >> there being any well ordering of the reals. The closest one can come is >> the H-riffic numbers. :) > > Hardly. The H-riffics are a simple countable subset of the reals. > Anyone mathematically inclined can come up with such a set. > You never paid enough attention to understand them. They cover the reals.
From: Tony Orlow on 3 Oct 2006 02:51 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? > The rules state that for each point in time, at 1/2^n seconds prior to > noon, 10 balls are added to the vase, and then the ball that was > previously inserted earlier than all the others is removed. (I.e., at > time 1/2^n, balls 10n+1 thru 10n+10 are added, and ball n is > removed.) > > Therefore the rules stipulate that balls are added and removed > at each moment 1/2^n sec before noon, for each n = 1,2,3,... . > > By assuming that there is some last moment when the vase is emptied, > or when the last ball is added, or whatever, you are assuming that > there is a largest n. That's your assumption, because it's not > mentioned in the rules. But it's an unwarranted assumption, by the > simple fact that there is no largest n (as you yourself have proclaimed > many times). > > [Personally, I think the problem is a bit simpler if only two balls > are added and one removed at each step. But, whatever.] > Sure, but that doesn't fix the problem with adding more than you're subtracting and coming up with an empty till.
From: Tony Orlow on 3 Oct 2006 02:55 Dik T. Winter wrote: > In article <1159649021.675137.307470(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > > So, again, no definition. Where did I not speak the truth? > > > > > > > > Here: "...because he never answered to questions about it". > > > > > > You gave on Usenet a definition of natural number in answer to my question > > > for it. I posted questions about that definition, and you never answered > > > them. So in what way is it a lie when I state that you never answered > > > them? > > > > I did not see an unanswered question. > > Apparently you did not see the questions at all. You stated as your definition > something like: > "numbers are in trichotomy with each other" > and I answered something like "so omega is a number". But I never did see an > answer. Is omega greater than any finite number? > > I asked for a mathematical definition of number. You never gave one (and > also your paper does not give any mathematical definition), your only > mathematical definition was about the trichotomy. That's an important question. Can you answer it? > > > Please repeat. > > If you do repeat. > > > By the way, please switch to the thread "Cantor > > Confusion" because this one has become too lengthy and, at home, I have > > only a slow internet access. So I am not able to follow this thread > > firmly. > > Oh. I did not know that slow internet access made long threads more > difficult to follow than short threads. I see no reason to shift the > subject. And certainly not to a subject for which a thread already does > exist. Yeah, that's kind of lame. > > > > > Most > > > > questions on the representation of a number are answered in my paper. > > > > > > The questions were about the definition you gave in Usenet. And I do not > > > ask about "representations", I ask for a *definition* of the concept > > > "number". A proper, mathematical, definition. > > > > In my paper which you read I quoted Peano and v. Neumann on the first > > page. Then, on page 2, I wrote "Of course this realization of 2 > > presupposes some a priori knowledge about 2. But here we are concerned > > with the mere realization." That is my point! > > A point, perhaps. But not a definition. > > > > > The definition of an object does not provide its existence. > > > > > > Indeed. But when it is *defined* as the limit of a sequence, and if that > > > limit exists, that means that the object does exist. > > > > The limit omega does not exist. > > Who insists that the limit is omega, except you? 0.111..., seen as a > decimal number exists (due to the definition) and is equal to 1/9. Seen > as a termary number it also exists, and is equal to 1/2. Seen as a > unary number it does not exist as a natural number. But in your sequence > I see it only as a sequence of digits, without interpretation, and as such > it does exist. That is what a number is to Wolfgang - a series of digits, or some other symbolic representation. > > > > > The cardinality of the set of prime numbers known today P(t) is as > > > > unbounded as the time variable t of today. > > > > > > If you talk like that you can not talk about "the set of known prime > > > numbers", because that is not a set, but a function of time. > > > > A set can be a function of time. > > No. A set is fixed. A function of time delivering sets is just that, a > function of time delivering sets. In the same way sin(x) is not a real, > it is a function of x delivering a real. > > > > > > For > > > each 't', the outcome is a specific set. A properly defined set does > > > not change over time. > > > > That is your definition, not mine. We have the same with numbers. Of > > course, every number is a constant. Nevertheless, variables can exist > > for numbers (which can, but need not, be defined as sets) and for sets. > > Variables do exist, but are not numbers. They represent numbers, and what are numbers, but representations of quantity? > > > "The letters X and Y in these expressions are variables; they stand > > for (denote) unspecified, arbitrary sets." Karel Hrbacek and Thomas > > Jech: "Introduction to set theory" Marcel Dekker Inc., New York, 1984, > > 2nd edition., p. 4. > > Yes, and so what? > > > > So when you talked about the set of known prime > > > numbers, I thought you were talking about the set of prime numbers known > > > at the time you wrote it, as I can give it no other interpretation. > > > > You fall back behind Cantor. He could. "ein in Ver?nderung Begriffenes > > Endliches, das also in jedem seiner actuellen Zust?nde eine endliche > > Gr??e hat." An example is the largest natural number you can > > imagine. Try it. There is always a larger one. > > What is the point? Graham's number is, eh, quite large. And there are > larger numbers. So, again, what is the point?
From: Virgil on 3 Oct 2006 02:56 In article <4521fdc2(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45215d2f(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Han de Bruijn wrote: > >>> stephen(a)nomail.com wrote: > > > >>>> 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. > >>> > >>> Han de Bruijn > >>> > >> Actually, that question is not ill-posed, and has a clear answer. The > >> vase will be empty, if there is any limit on the number of balls, and > >> balls can be removed before more balls are added, but it is not the > >> original problem, which states clearly that ten balls are inserted, > >> before each one that is removed. That's the salient property of the > >> gedanken. Any other scheme, such as labeling the balls and applying > >> transfinitology, violates this basic sequential property, and so is a ruse. > > > > One can pose any gedanken one likes. > > > > If TO does not like to be able to tell one ball from another, he does > > not have to play the game, but he should not ever try to pull that in > > games of pool or billiards. > > If distinguishing balls gives a less exact answer, and a nonsensical one > to boot, then that attention can be judged to be ill spent, and not > contributing to a solution at all. TO's dislike does not constitute inexactness. Distinguishing the balls, as the rules require, gives an absolutely exact answer, but one TO does not like. It is at least equally nonsensical for TO to argue that when the ball for every number has been removed and no unnumbered balls were inserted that there are still some presumably unnumbered balls left. As they have never been inserted into the vase, they must have been born there. Are they the illegitimate offspring of some of the numbered balls, TO?
From: Tony Orlow on 3 Oct 2006 03:01
Virgil wrote: > In article <45210227(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <45205fa9(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <45203919(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>> >>>>> >>>>>>> Since ordinals are, by definition, well ordered, they cannot contain any >>>>>>> endlessly decreasing sequences, which TO's models require. >>>>>> Neither can the reals. >>>>> How about the set of negative integers? >>>>> How is that not an endlessly decreasing sequence of reals? >>>> The origin is at a finite location. Order starts from the bottom, if >>>> "decreasing" has any meaning. >>> The set of negative integers has no "bottom". >>> >>> TO seems to be changing his tune when it is used against him. >>> >>> According to TO every set of numbers has a natural order, and it is >>> within that natural order that we must view it, but now he wants to >>> reject the natural order because it runs counter to another of his >>> claims. >>> >>> TO blows hot and cold with the same breath. >> I'm saying the if you iterate the negative integers starting at 0, in >> that order, there is no infinite descending sequence. > > > But that is not their "natural" order, and TO elsewhere insists that we > follow natural orderings. From Dik Winter, elsewhere in this thread: "But that has ordinal k + omega, not -k + omega. A set of k negative numbers has ordinal k; not -k." |