Cantor Confusion
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From: Andy Smith on 3 Feb 2007 18:46 > David Marcus > wrote: > >>Dave Seaman wrote: >>> On Fri, 02 Feb 2007 20:29:03 GMT, Andy Smith wrote: >>> > Andy Smith <Andy(a)phoenixsystems.co.uk> writes >>> > (snip everything else) >>> >>> > At root I think my problem comes down to achieving a suitably Zen-like >>> > perspective on the following apparently incompatible statements: >>> >>> > 1) The real line is made up of an ordered and infinite set of points, >>> > and is connected. >>> >>> > 2) No point on the real line has an adjacent point. >>> >>> I don't understand why you think those two statements are incompatible. >>> If any point on the real line actually *had* an adjacent point, then the >>> line would be disconnected precisely at the gap between those two points. >>> Hence, connectedness is incompatible with the existence of adjacent >>> points. >> >>I suppose he is thinking of points as having size, e.g., like little >>marbles. Of course, they aren't like that. > No, I think of points as 0-dimensional, with no size. They must necessarily have zero size, otherwise you couldn't have an infinite number of points in a finite distance. The issue was whether you could actually "cover" the line with points. But I can see that my Euclidean view of a line and dimensionality of points, lines and planes is not sustainable, and I can't see anything wrong with your definition of the line, other than my mis-intuition; let's close this discussion now, with thanks. -- Andy Smith
From: David Marcus on 3 Feb 2007 20:21 Andy Smith wrote: > Lester Zick <dontbother(a)nowhere.net> writes > >On 2 Feb 2007 13:31:10 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > > >>No Zen-like perspective is required. Knowing the axioms and > >>defintions, though, does help. > > > >As does knowing Zen. > > I read "Zen and the Art of Motorcycle Maintenance" once, which I recall > was mildly interesting & entertaining. > > In Western society I think that it is common parlance to describe > something as "Zen-like" to imply either, that there exists a deep > resolution of some apparently irreconcilable statements, or that > consideration of some suitably impossible conundrum may allow some > enlightenment on a related problem. > > I meant no disrespect to any religious beliefs that you may hold ... An apt way of describing Lester's knowledge of math. -- David Marcus
From: Dik T. Winter on 3 Feb 2007 21:36 In article <1170470698.824513.309540(a)m58g2000cwm.googlegroups.com> cbrown(a)cbrownsystems.com writes: > On Feb 2, 5:12 pm, Carsten Schultz <cars...(a)codimi.de> wrote: .... > > From that it follows that there are exactly four sets with four > > elements, since these are the elements of 4. It also follows that there > > is only one set with four elements, namely four. So 4=1. You should > > write a book about this. > > Dik is even reviewing it, I think. I am. I am now halfway chapter 8 and only found one serious error (in chapter 7). And one place where I have serious doubts (in chapter 8, but I have to look thoroughly at that). But until this point it is an excellent review about the history about the thinking about the infinite. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 3 Feb 2007 22:18 In article <MPG.202ec75a1e9729c2989c32(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article <MPG.202e854deaa10136989c30(a)news.rcn.com>, > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > Virgil wrote: > > > > In article <45c31c0b$0$97214$892e7fe2(a)authen.yellow.readfreenews.net>, > > > > Franziska Neugebauer <Franziska-Neugebauer(a)neugeb.dnsalias.net> wrote: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > On 2 Feb., 02:42, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > > > > >> However, when we construct the path: > > > > > >> p = {n_1} U {n_1, n_2} U {n_1, n_2, n_3}, ... > > > > > >> we get as path: > > > > > >> {n_1, n_2, n_3, n_4, ...} > > > > > >> the path length in this case is *not* a natural number. It is the > > > > > >> cardinality of N. > > > > > > > > > > > > The pathlength is not a natural number but it is a number (by > > > > > > definition), namely omega. You see that there is no infinite set N > > > > > > without an infinte number in it. > > > > > > > > > > As omega is not member of N I don't see that. > > > > > > > > Also, every natural is either even or odd but not both. > > > > is omega even or odd? > > > > > > You are missing a WM axiom: Path lengths are natural numbers (even for > > > infinite paths). > > > > On the contrary, I don't miss it a bit! > > But, if you assume it, you can prove so many more things! if one asssumes 1 = 2, one can prove all sorts of curious things, but that does not justify assuming 1 = 2.
From: mueckenh on 4 Feb 2007 04:38
On 2 Feb., 12:10, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > > On 2 Feb., 02:42, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > [...] > >> However, when we construct the path: > >> p = {n_1} U {n_1, n_2} U {n_1, n_2, n_3}, ... > >> we get as path: > >> {n_1, n_2, n_3, n_4, ...} > >> the path length in this case is *not* a natural number. It is the > >> cardinality of N. > > > The pathlength is not a natural number but it is a number (by > > definition), namely omega. You see that there is no infinite set N > > without an infinte number in it. > > As omega is not member of N I don't see that. The total pathlength is the union of all single pathlengths. The total pathlength cannot be omega without omega being a pathlengt (in the union). But omega cannot be in the union of finite pathlengths. Hence there is no actually infinite pathlength. "Infinite pathlength" means only that every finite pathlength is surpassed by another finite pathlength. Regards, WM |