Cantor Confusion
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From: David Marcus on 2 Feb 2007 16:51 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. Do you have the same problem with the following? 1) The real numbers consist of an infinite number of numbers that are ordered and connected. 2) No real number has an adjacent real number. 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. -- David Marcus
From: Lester Zick on 2 Feb 2007 19:10 On Fri, 2 Feb 2007 14:09:42 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Andy Smith wrote: >> In message <epvbqo$bjl$1(a)mailhub227.itcs.purdue.edu>, Dave Seaman >> <dseaman(a)no.such.host> writes [. . .] >> 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. So we're doing parables instead of math now? > And, you passed through each >point at some time. Or you didn't. Might we dispense with the metaphors already? > And, the time you were at a point is the same as the >distance you had travelled to get to that point. Oh gloria!! We are indeed saved. > And, we can measure >time (and distance) using real numbers. And what are these magic things you call real numbers pray tell? > So, if you think lines aren't >made up of points, then time isn't made up of instants. So if time isn't made up of instants lines aren't made up of points? Rather a curious concept I must say but correct as far as it goes. ~v~~
From: Lester Zick on 2 Feb 2007 19:12 On Fri, 2 Feb 2007 16:49:09 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> 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. Of course they aren't. Infintesimals have size not points. Your brain is like little marbles. ~v~~
From: Lester Zick on 2 Feb 2007 19:13 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. ~v~~
From: Lester Zick on 2 Feb 2007 19:22
On Fri, 2 Feb 2007 16:51:39 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >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. > >Do you have the same problem with the following? > >1) The real numbers consist of an infinite number of numbers that are >ordered and connected. An "infinite number of numbers"? And what exactly does that mean when it's at home? >2) No real number has an adjacent real number. It doesn't? And why perchance not pray tell? The last I heard from Bob Kolker the real number line is densely pointy.I mean unless you really mean to indicate the real number line is sparsely pointy in which case you really need to take it up with Bob since I, like mathematicians, have no preference in the matter because I use lines to define points. >Of course, the real problem is that you are relying on your intuition. Whereas you prefer to rely on someone elses intuition. >The first thing a mathematician learns is to adjust their intuition to >match the facts. Which facts exactly did you have in mind? True facts or just any old facts? ~v~~ |