Cantor Confusion
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From: David R Tribble on 3 Feb 2007 12:55 David Marcus wrote: > Of course, the real problem is that you are relying on your intuition. > The first thing a mathematician learns is to adjust their intuition to > match the facts. True of mathematicians, logicians, scientists, and realists, all of whom deal with some kind of demonstrably truths derived from simpler truths, whether they are abstract or physically concrete. On the other hand, those that do not make this adjustment are generally of the more philosophical or religious bent, who deal with beliefs and rules only loosely based on (or not based on at all) provable logical or physical truths. Thus it's pretty accurate to say that mathematical cranks are more religious in their fervent beliefs than true mathematicians. Indeed, mathematicians can be downright agnostic about many things (e.g., the CH).
From: Virgil on 3 Feb 2007 14:31 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!
From: David Marcus on 3 Feb 2007 16:23 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! -- David Marcus
From: Andy Smith on 3 Feb 2007 17:57 Lester Zick <dontbother(a)nowhere.net> writes >On 2 Feb 2007 13:31:10 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > >>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. > >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 ... -- Andy Smith
From: Andy Smith on 3 Feb 2007 18:11
><DavidMarcus(a)alumdotmit.edu> wrote: > > Your post appears to have been deleted off the NG server, or else my NG reader can't cope with this thread. Piecing it together: >>> You just start with points and define the real line as the set of all of >>> them. I have an image of a line as a continuous thing of point width, >>> and it is trying to marry up the perception of continuity with the set >>> of real points that is difficult for me. >> >>Suppose at time zero you start walking in a straight line at constant >>speed. At any time, you are at some point. > >>And, you passed through each >>point at some time. > >>And, the time you were at a point is the same as the >>distance you had travelled to get to that point. > >>And, we can measure >>time (and distance) using real numbers. > >>So, if you think lines aren't >>made up of points, then time isn't made up of instants. > OK, I understand that, and that is all self-consistent. Both distance and time along conceptual axes are equivalent, but (my naive view) would have been that both were 1-dimensional entities, and that time was no more made up of connected but non-adjacent instants than the line is made up of connected but non-adjacent points. -- Andy Smith |