Cantor Confusion
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From: Andy Smith on 1 Feb 2007 15:22 Dave Seaman <dseaman(a)no.such.host> writes (snip) > >> 1) Is there a set with cardinality greater than N but less than R ? > >That question is known to be undecideable from the axioms of ZFC. Google >for "Continuum Hypothesis". > I had previously imagined that the "Continuum Hypothesis" related to whether the set of reals (as 0 dimensional points, each with no neighbouring point) could cover the line, with dimension 1 (Which seemed like a good question). But at least I now understand the significance of the "Continuum Hypothesis", thanks. -- Andy Smith
From: Dave Seaman on 1 Feb 2007 15:41 On Thu, 01 Feb 2007 20:22:27 GMT, Andy Smith wrote: > Dave Seaman <dseaman(a)no.such.host> writes > (snip) >> >>> 1) Is there a set with cardinality greater than N but less than R ? >> >>That question is known to be undecideable from the axioms of ZFC. Google >>for "Continuum Hypothesis". >> > I had previously imagined that the "Continuum Hypothesis" related to > whether the set of reals (as 0 dimensional points, each with no > neighbouring point) could cover the line, with dimension 1 (Which seemed > like a good question). You are asking whether R and "the real line" are the same set. In order for that question to have a nontrivial answer, you must have some meaning in mind for "the real line" other than R. What is your definition? > But at least I now understand the significance of > the "Continuum Hypothesis", thanks. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: Andy Smith on 1 Feb 2007 16:52 Dave Seaman <dseaman(a)no.such.host> writes (snip) >>> >> I had previously imagined that the "Continuum Hypothesis" related to >> whether the set of reals (as 0 dimensional points, each with no >> neighbouring point) could cover the line, with dimension 1 (Which seemed >> like a good question). > >You are asking whether R and "the real line" are the same set. In order >for that question to have a nontrivial answer, you must have some meaning >in mind for "the real line" other than R. What is your definition? > Well until recently my naive view was that points and lines were different entities, and you might as well ask how many metres in an acre as how many points in a line. I would have said (in your terminology) that there is clearly an injection from the reals into the line, but that you can never cover it. But I doubt now if that is true, or if the question is a sensible one to start with. I am exercised by the idea of space filling curves, which is the same issue, covering something by something of a lower classical dimension, and so maybe the recursive fractal definition of reals does cover my concept of line. I asked some time before on the other thread, and didn't really get a straight answer from DM, about a fractal scenario where case 1: we start of with the interval 0-1. On each iteration, each interval is divided in two, and we associate a real number with the mid-point of each section. After an infinite number of iterations, all the reals in [0,1] are covered, no problem. Case 2: a variant on that, we have colours black and white, and start off with the interval 0-1 painted white. On each iteration, each interval is subdivided, with the first sub-segment of each sub-interval painted black if its parent interval was white, or vice-versa. Then, after an infinite number of iterations, is each real point painted black or white? And, if we decide to delete all white points after an infinite number of iterations, would that make any difference? Or is that all completely irrelevant? -- Andy Smith
From: Lester Zick on 1 Feb 2007 17:13 On Thu, 1 Feb 2007 13:24:10 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> On 30 Jan., 20:25, Virgil <vir...(a)comcast.net> wrote: >> > In article <1170156979.321128.184...(a)j27g2000cwj.googlegroups.com>, >> >> > > I can't tell whether the union of all lengths is finite or not. >> > >> > As the union of all lengths is obviously equivalent to the union of all >> > finite ordinals and the union of all finite ordinals is clearly not >> > finite, WM can't tell much. >> >> The union of all lengths is not the sum of all lengths, which, >> according to your standpoint might be infinite. But the union of all >> lengths is a length. You may say that it is infinite or not. If it is >> infinite, then it corresponds to an infinite natural number. If it is >> not infinite, then the set N does not exist other than as a >> potentially infinite (= finite but unbounded) set. > >It is amazing what you can write if you fail to define your terms. If we >read it quickly, it sounds vaguely like a logical argument. Of course, >it is actually nonsense. Did you have to practice for a long time to >develop this technique? Well no, not exactly. I had to study quite a while to understand that definitions are only abbreviations but once understood the abbreviations came quite easily. That way I could concentrate my efforts on making my arguments sound logical when of course they were like your definitions actually nonsense. ~v~~
From: David Marcus on 1 Feb 2007 17:56
Andy Smith wrote: > Dave Seaman <dseaman(a)no.such.host> writes > > >You are asking whether R and "the real line" are the same set. In order > >for that question to have a nontrivial answer, you must have some meaning > >in mind for "the real line" other than R. What is your definition? > > Well until recently my naive view was that points and lines were > different entities, and you might as well ask how many metres in an acre > as how many points in a line. I would have said (in your terminology) > that there is clearly an injection from the reals into the line, but > that you can never cover it. But I doubt now if that is true, or if the > question is a sensible one to start with. You are a hundred or more years behind the times. In the old days, geometry and analysis were separate. I suppose Descartes gets credit for starting the trend to merging them. These days, we find it simplest to consider numbers to be more fundamental and to define geometric objects in terms of numbers. In particular, the Euclidean plane is defined to be R^2, i.e., ordered pairs of real numbers. This makes the x (or y) axis the same thing as R. Using this definition, we can recover all of Euclidean geometry as theorems without having to introduce any new axioms. This makes life simple. > I am exercised by the idea of space filling curves, which is the same > issue, covering something by something of a lower classical dimension, > and so maybe the recursive fractal definition of reals does cover my > concept of line. No. This is a different problem. Even Euclid treated lines as consisting of points and planes as consisting of lines. A space-filling curve is a is a continuous function from [0,1] to [0,1]^2 that is onto. The surprising feature is that the function is continuous. > I asked some time before on the other thread, and didn't really get a > straight answer from DM, If you don't get a "straight answer", maybe you didn't ask a straight question. Your question was about changing the space-filling curve construction, but it wasn't clear how you wanted to change it. > about a fractal scenario where case 1: we > start of with the interval 0-1. On each iteration, each interval is > divided in two, and we associate a real number with the mid-point of > each section. After an infinite number of iterations, all the reals in > [0,1] are covered, no problem. Do you mean that at stage n+1, we divide all the intervals from stage n in half? So, at each stage we have twice as many intervals as before? If we do this for all natural numbers n, we only get a countable number of intervals (all together), so the mid-points of the intervals most certainly do not include all real numbers in [0,1]. The only numbers you get are numbers with a finite binary expansion. > Case 2: a variant on that, we have colours black and white, and start > off with the interval 0-1 painted white. On each iteration, each > interval is subdivided, with the first sub-segment of each sub-interval > painted black if its parent interval was white, or vice-versa. Then, > after an infinite number of iterations, is each real point painted black > or white? If by "after an infinite number of iterations", you mean the limit of the color, then for almost all points, there is no limit. The sequence of colors keeps alternating white, black, white, etc. The only points that have a limit are those with a finite binary expansion. > And, if we decide to delete all white points after an infinite > number of iterations, would that make any difference? Make any difference to what? There are only a countable number of white points and a countable number of black points. > Or is that all completely irrelevant? Irrelevant to what? -- David Marcus |