An uncountable countable set
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From: Franziska Neugebauer on 19 Aug 2006 12:10 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). > 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). >> 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. -- xyz
From: Tony Orlow on 19 Aug 2006 12:11 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 most people have any doubt that I think, whether they agree with my lack of orthodoxy or not. Have a nice day. TO
From: Tony Orlow on 19 Aug 2006 12:25 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, is square at every step, and yet, has no columns with infinitely many 1's in it? We made a deal, yes? Your turn. :) > > 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? Isn't The width the same as the >> height of the square, as drawn? I say the set, as defined, is the same >> size as its maximum element. >> >> :) > TO
From: Tony Orlow on 19 Aug 2006 12:33 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. >>> You just proved it to be true. >> You may "just" as well prove the opposite. Please do so. >> >>> 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. > There are infinitely many differences, if we consider infinitely many > numbers. > 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. > >> Do you agree that P ~ omega? > > 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. > >> 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. > > Regards, WM > Hi WM. I agree entirely with your analysis regarding the interdependence of the existence of the infinite set of naturals and the existence of naturals of infinite value. This leads to the conclusion that the set of finite naturals is finite, though unbounded, since as you say, there cannot be an infinite number of differences of 1 where no two elements are infinitely different. In order for an infinite difference to occur between two natural, one must have an infinite value. However, I wonder whether you would ever consider the existence of infinite natural numbers. After all, your argument says that either you have a finite set OR you have infinite values in the set. Is the second option objectionable for you? :) TO
From: Tony Orlow on 19 Aug 2006 12:36
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 refuse to understand what "to exist" means. In no case infinitely > many differeces of 1 can exist unless infinitely many diferences of 1 > do exist. But that means an infinite size. > > Regards, WM. > Well, here you answer my question just posed to you in my last post. There is no infinite number, in your opinion. But, how do you reconcile this with the infinity of reals in any unit interval? If, between any two distinguishable reals there is a third distinguishable from the first two, then does this not imply that we can subdivide the unit interval infinitely, yielding an infinite set of interval endpoints, aka reals? Isn't the set of reals in (0,1] infinite, and how do you characterize this number? Thanks, TO |