An uncountable countable set
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From: Tony Orlow on 19 Aug 2006 13:21 Franziska Neugebauer wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > >> Franziska Neugebauer schrieb: >>>>>> Therefore there are not infinitely many difference[s] of 1 >>>>>> between natural numbers. >>>>> Your consequent is proven false (see below). Therefore your >>>>> implication is false, too. >>>> You are in error. >>> Where precisely is the error? >> The assertion that infinitely many differences of 1 can be provided by >> finite natural numbers. > > So you claim my proof is wrong. Where exactly? > >>>> You just proved it to be true. >>> You may "just" as well prove the opposite. Please do so. > > Don't you want to? > >>>> The set of natural numbers (i.e., finite numbers n, i.e., numbers >>>> with finitely many differences of 1 between 1 and n) does not yield >>>> infinitely many differences of 1. >>> This is a reiteration not a proof. >> There are not infinitely many differences, if we consider the finity >> of each number. > > Proof? > >> There are infinitely many differences, if we consider infinitely many >> numbers. > > There _are_ infinitely many numbers n e omega and therefore there are > "infinitely many differences of 1" (as proven). Well, Franziska, if there are infinitely many differences of 1 between any two elements of N then one or the other must be infinite. However, given that all m and n e N are finite, m-n is also finite, and there is no infinite distance between any two elements, and the set has finite range and size, given the unit differences between consecutive elements. > >> Therefore one of the assumptions is wrong: Either there are not >> infinitely many numbers or there is at least one number which is >> infinite by size. > > There are infinitely many numbers n e omega _and_ there is not a single > number which is "infinite by size". This is fact and hence your claim > is false (and unproven, too). It is only a fact given one's definition of the infinitude of a set. By the accepted set theoretical system of transfinite cardinalities, sure, you're correct. But it doesn't jibe with more traditional, more tried-and-true quantitative modes of thought. Formulas go out the window with transfinite set theory, elements are added without increasing the set, and computation fails. Why else would there be such fervent objections to Cantorian hocus pocus? See? Even avid infinitrons like myself and devout finitists like WM see the same problem. Once we get Cantor out of the way, then WM and I will have to duke it out, eh? ;) TO > >>> Do you agree that P ~ omega? >> P has the same cardinality as |N. > > So P ~ omega. > >> But either there are not infinitely many numbers or there is at least >> one number which is infinite by size. > > Why (please prove!)? Do you want to ignore facts? > >>> Do you agree that card (omega) /e omega? >> card (omega) = aleph_0 = the number of natural numbers and that cannot >> be infinite, as we have seen. > > "Cannot"? > > "We" have not (yet) _"seen"_ this. It is fact that card(omega) /e omega > (not *in* omega) and hence not finite. Do you like to argue against > facts? > > F. N.
From: Tony Orlow on 19 Aug 2006 13:23 Virgil wrote: > In article <ec7d7r$ath$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Virgil wrote: >>> In article <ec5k1i$4gr$2(a)ruby.cit.cornell.edu>, >>> Tony Orlow <aeo6(a)cornell.edu> wrote: >>> >>> >>>> By the way, hello! Nice to see you all again, I think. >>> >>> Does Tony actually think? >> Nice to see you too, Virgil. I don't think > > I thought not. Well, that was lame. You can do better. (I hope) But, I set you up for a "number 2" joke, just to ease you back into the swing of things. Oh yeah, I'm the one who's been away. But, were you ever really here? Hmmmm.... ;) TO
From: imaginatorium on 19 Aug 2006 13:40 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> David R Tribble wrote: > >>> mueckenh wrote: > >>>> Learn: Infinitely many stairs require infinite height. But that it not > >>>> admitted in mathematics. There have to be infinitely many stairs which > >>>> give an infinitely long staircase ... > >>> Yes, infinitely many finite-sized steps produce a staircase of infinite > >>> height. That is admitted in mathematics. > >> All too often "admitted" that those infinities are equal. > >> > >>>> ... but this staircase is allegedly of finite height. > >>> No, the staircase is infinite in height, but each step is finite in > >>> height. The distance between any two steps is finite, but the total > >>> height of all the steps together (the entire staircase) is infinite. > >>> > >> So, the greatest possible value that any natural number can reach, in > >> height, is infinite? > > > > Hello Tony! > > Hi Brian!! How's tricks? > > I'll assume this is a question: > > > > Q1: "Is the greatest possible value that any natural number can reach > > infinite in > > height?" > > Okay, you can rephrase it like that, sure. It had a question mark anyway. > > > > > I'll tell you the answer to Q1 when you tell me the answer to Q2... > > Okay. > > > > > You know that the sharpest corner of any triangle is less than 90 > > degrees; the sharpest corner of an almost regular pentagon is more than > > 90 degrees. > > Sure. > > > > > Q2: "Is the sharpest corner of a circle less than or greater than 90 > > degrees?" > > So, if I answer this question, you will answer mine? Deal. > > A circle is a regular polygon with an infinite number of infinitesimal > sides. The formula for the angle of the vertices of a regular polygon of > n sides is 2*pi*(1/2-1/n). The limit of this formula as n->oo is pi. > Each vertex of the circle has an angle of pi. That is, the change in > angle at each vertex of the circle is infinitesimal, which makes it zero > in the standard mathematical world. So, the angle at each vertex of the > circle is greater than 90 degrees. > > Now, it's your turn. With the addition of each natural in turn, the > width of the unary list is precisely equal at all times to its height, > each time incrementing equal values as elements are enumerated, > maintaining the squareness of the list, inductively demonstrable. So, > how is it that you have such a list which has infinitely many > successively greater columns [[shouldn't this be 'rows'?]], is square > at every step, and yet, has no columns [[rows?]] with infinitely many 1's in it? The final drive shaft on our prewar Morris Minor rotated clockwise (looking backwards). I've spent a lot of time wondering about the direction of rotation of the drive shaft in a dog. Now I know. It rotates in the chocolate direction. (It is incontrovertible that for every drive shaft in every dog, there is no rotation in any direction other than chocolate.) Well, it's the same, isn't it. When you arrive at this "list with infinitely many rows, the bottom one is of length 42. There is no bottom one of any length other than 42, and every row must have a length, therefore it is 42. On the other hand, I admit it's rather easy to prove that the bottom row in a bottomless list of rows is actually infinite in length, green in colour, and weighs just 20g. Oh, is that what you wanted? The last number in an unending sequence of numbers is infinite? Sure. And pink. And yellow. And colourless. > We made a deal, yes? Your turn. :) I note that your "circle" response makes pretty heavy use of undefined terms (f-words and i-words), and relies on the properties of clearly nonexistent objects. Well, good luck getting your paper accepted somewhere. Brian Chandler http://imaginatorium.org @ nothing.changed.i.c > > Certainly you claim that the width of this square, > >> the count of all such values, is infinite. But are the values > >> themselves, even potentially, infinite? What does it mean for a single particular value to be "potentially" whatever?
From: Virgil on 19 Aug 2006 13:47 In article <1155998489.106225.154260(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Where precisely is the error? > > The assertion that infinitely many differences of 1 can be provided by > finite natural numbers. There is, in fact, only one difference of 1, but there are infintely many pairs of naturals that produce that difference. > There are not infinitely many differences, if we consider the finity of > each number. > There are infinitely many differences, if we consider infinitely many > numbers. There are as many differences, n - 0, as there are naturals n \in N. > Therefore one of the assumptions is wrong: Either there are not > infinitely many numbers or there is at least one number which is > infinite by size. False dichtomy, it ignores the actual case: that there are infinitely many finite numbers. > > P has the same cardinality as |N. But either there are not infinitely > many numbers or there is at least one number which is infinite by size. That is Tony Orlow's delusion, too. > > > Do you agree that card (omega) /e omega? > > card (omega) = aleph_0 = the number of natural numbers and that cannot > be infinite, as we have seen. WE have not seen any such thing. It is fairly easy to see that if there is no injection from the set of naturals to any proper subset, then there must be a largest natural. And if there is such an injection, then the set is Dedekind infinite. So that "Mueckenh" must choose between infinitely many naturals or a largest natural. Well which is it "Mueckenh"?
From: Virgil on 19 Aug 2006 13:48
In article <1155998568.859179.65830(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > You have agreed to that this limit does not exist ("There is no L in > > N"). So the sum is at least _not_ _finite_ (not in omega). > > > There is no L, nowhere. It is wrong to say that L is in |N and it is > wrong to say that L is larger than any n e |N. Actual infinity does not > exist! I told you that already several times. But "Mueckenh" has not been able to give us any compelling reason to believe him, and we have compelling reasons not to. |