Cantor Confusion
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From: Dave Seaman on 2 Feb 2007 15:42 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. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Virgil on 2 Feb 2007 15:53 In article <1170413742.648825.136900(a)l53g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 2 Feb., 02:42, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1170348637.768496.237...(a)a75g2000cwd.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > > > So the union of finite chains of nodes like the following > > > 0 > > > 12 > > > 6543 > > > does not contain an infinite chain of nodes of the form > > > 0 > > > 12 > > > 6543 > > > 789... > > > ??? > > > > I did not state that. Pray try to read what I wrote, rather than giving > > your own interpretation to it. > > > > > Frankly, I do not see much difference to: The union of finite paths > > > contains an infinite path. > > > > But that is true. > > So you say: > The union of finite trees contains an infinite tree. > The union of finite chains contains an infinite chain. > The union of finite paths contains an infinite path. > > > What I state again, again and again, but what you > > misread each and every time is: "the union of sets of finite paths > > does not contain an infinite path". > > > So you say the union of finite paths > p(0) U p(1) U p(2) U ... = {0} U {0,0} U {0,0,0} U ... > contains the infinite path p(oo) = {0,0,0,...}. > > But the union of finite sets of finite paths > {p(0)} U {p(1),q(1)} U {p(2), q(2), r(2),s(2)} U ... > does not contain the union of finite paths > p(0) U p(1) U p(2) U ... = {0} U {0,0} U {0,0,0} U ... If p(n) = {node_0, node_1, node_2,...node_n} = { node_k : k in N and k <= n} then Union_{n in N} p(n) = {node_0, node_1, node_2, ...} = { node_n : n in N} is an endless sequence of nodes, which is a path, but Union_{n in N} {p(n)} = { p(0), p(1), p(2),...} = { p(n) : n in N } is an endless sequence of paths, which is not a path. Note that { node_n : n in N} != { p(n) : n in N } and { node_n : n in N} is a path but { p(n) : n in N } is a set of paths and not a path.
From: Virgil on 2 Feb 2007 15:55 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. > > F. N. Also, every natural is either even or odd but not both. is omega even or odd?
From: MoeBlee on 2 Feb 2007 16:31 On Feb 2, 12:29 pm, Andy Smith <A...(a)phoenixsystems.co.uk> wrote: > At root I think my problem comes down to achieving a suitably Zen-like > perspective No Zen-like perspective is required. Knowing the axioms and defintions, though, does help. MoeBlee
From: David Marcus on 2 Feb 2007 16:49
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. -- David Marcus |