Cantor Confusion
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From: Lester Zick on 26 Nov 2006 15:11 On Sun, 26 Nov 2006 03:32:30 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <962hm2tg9dqirfmerue74rasendvcggoll(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > On Sat, 25 Nov 2006 03:12:20 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: >... > > >You completely misunderstand what I wrote. I *start* with an Euclidean > > >plane without measure (i.e. distance function). With that we can at > > >most define a rectangle. Neither a square, nor a circle. > > > > Actually I think I understand you very well. How is it exactly you > > start with a Euclidean plane? > >The plane as defined by Euclides. With all his axioms (or postulates). > > The fact that you start without a > > distance measure doesn't allow you to start with a plane Euclidean or > > otherwise. > >Why not? Euclides did not have a distance function either. He didn't try to make an argument for square circles either. You do. > > If you begin by assuming this you wind up by assuming that > > and pretty soon you find yourself assuming what you were supposed to > > demonstrate in the first place, that square circles are conceivable. > >Well the latter was not an assumption. There was an explicit question. But no demonstrated answer. You just claimed square circles are conceivable according to a metric you can't demonstrate in the same Euclidean terms you claim to rely on. > > > I do not use Euclidean metric at all. Where, above, do I use Euclidean > > > metric? > > > > I suspect we're using the phrase "Euclidean metric" in different ways. > > When I use the term I'm referring not just to measures of distance but > > to all definitive characteristics which go into definitions of such > > things as dimensionality and geometric figures in addition to distance > > measures. > >In that case you are using quite non-standard terminology. Modern mathematikers use quite non standard terminology to refer to an imaginary real number line. If it doesn't bother them no reason it should bother me. > > For example I see no definition of yours for "plane" which I > > think would be impossible to define without a Euclidean metric. > >See Euclides' postulates. > > > On the > > other hand if all you're describing are variable measures of distance > > then you'd have to explain how you obtain those measures without an > > underlying Euclidean metric and corresponding assumptions. You can't > > just assume them as modern mathematikers are wont to do. > >You read more in the word "metric" than is in there. Not at all. Even in the restricted sense of "distance measure" a term like "metric" has to be self consistent with underlying assumptions of measure and what is measured and how. > > > > For example is it possible to define plane squares with non Euclidean > > > > metrics? > > > > > >Of course. > > > > Well then let's see some definitions for planes, circles, and squares > > which don't explicitly or implicitly rely on Euclidean assumptions. > >Now you require that without assumptions rather than metrics... That >is something completely different. Let us recap the Euclidean postulates: >(1) Two points determine a line >(2) Any line segment can be extended in a straight line as far as desired, > in either direction >(3) Given any length and any point, a circle can be drawn having the length > as radius and that point as center >(4) All right angles are congruent >(5) The parallel postulate >Now, that is precisely what I did use (after translation to analytical >geometry). The problem is you have a host of undefined terms for things like points, lines, length, circles, right angles, etc.If Euclid understood these things one way and you understand and employ them another they're not Euclidean assumptions at all just revisionist private definitions. To justify your claim that square circles are conceivable you also claim a metric you don't demonstrate on the basis of Euclidean assumptions you claim to rely on. > > > > And can we define right angles without the parallel postulate > > > > people simply ellide when operating with non Euclidean geometries? No. > > > > > >I think you can. But I used Euclidean geometry above. > > > > But the problem here is what kind of definition for squares doesn't > > rely on straight lines, right angles, planes, and so on? > >See above, postulate (4). > > > > > When I > > > > ask a question such as "why are square circles unimaginable" the > > > > defining metric for "squares" and "circles" is Euclidean and not > > > > Manhattan and I don't expect answers that if we look at Euclidean > > > > figures through some other metric we'll find they are imaginable. > > > > > >In that case you should use better formulations in your questions. > > > > And you should use better formulations in your answers. > >How could I give a better formulation than stating that in the Manhattan >measure it is possible? Your stating it has nothing to do with whether it's true. > I did state that explicitly, while you did not >state that you did use Eucledian measure. No reason I should if the words themselves rely on the same Euclidean assumptions you claim to rely on. > > > And > > >when I follow-up to point out that in the Manhattan measure all circles > > >are squares (but not the other way around) you should state that your > > >formulation was insufficient. > > > > Except that your answer relies on non Euclidean assumptions that > > circles are squares. > >No. I show that with a particular measure all circles are squares. There >is no assumption. So you say but so you don't demonstrate. What you say is nothing more than a claim. You might just as well reply "square circles are conceivable" and let it go at that. You haven't explained why square circles are imaginable except something called the Manhattan metric says so. Anyone could do as much just by saying "circles are squares" without all the metric nonsense. > > If I ask why one sided quadrangles are > > unimaginable and you reply that they aren't if you start counting from > > four would you consider your answer responsive to the question asked? > >You are using a few words that you did not define, and again using some >assumptions. Quadrangle I can understand. What is one-sided? Does it >mean that the top side is different from the bottom side? In that case >I can show you quadrangles on a Moebius strip that have only one side. >On the other hand, I think you are referring to the edges. In that case, >in topology a connected set (as a quadrangle is) has a single boundary. So you wind up with a four sided figure with one side? I don't see that you're taking your words or mine seriously. How is it you get from one side to another regardless of topology and how do you get past the points of intersection? > > All you're doing is answering a question that wasn't asked in terms > > employed by the original question. > >The terms employed by the original question did not refer to Euclidean >metric. So my answer was valid. Terms like "circle" and "square" don't refer to Euclidean metric? Okay then. Obviously this discussion is going nowhere. We might just as well move on to a discussion as to why points are lines and cats are dogs since it doesn't appear you have any ability to take these things seriously. > > I can make up private definitions > > just like everyone else does but that doesn't make definitions true. > >A definition can not be false. Another interesting and rather self serving claim from someone who obviously prefers false definitions to true definitions. > It can be the case that there is nothing >(within the current context) that satisfies the definition, but it is >still not false. So is it the case that definitions cannot be self contradictory or that self contradiction is not false? Hell I don't mind if modern mathematikers need to make up another word for "false" so they can feel useful for a change. But I think we need to be clear on what we're talking about. I mean if you think definitions cannot be self contradictory what is it you imagine goes on in the context of self contradictory predicates in definitions? Is there a variable logic for different predicates, a private definition for self contradiction that makes definitions exempt from being false? > > In fact my personal favorite private definition for distance metrics > > is one I made up for the real number line which runs 1, 2, e, 3, pi, > > 4, 5, . . but I don't try to pretend that when I'm trying to analyze > > real numbers that that is a true definition. > >That is not enough for a definition. Sure it is. I just said so and you just said that definitions cannot be false. > > Besides as far as I can tell you still haven't answered my question as > > to whether there are any curves at all with the Manhattan measure. In > > fact I can't even find it reprinted above. > >Pray define "curve". Don't need to. For purposes of the present discussion it is enough to understand that circles are curves and squares aren't. If you maintain otherwise I'd like to know how and whether there are other curved figures besides circles in the Manhattan metric which are squares. Quite frankly I don't understand how it is you can pretend to say anything of mathematical significance without understanding the difference between curves and straight lines. Do you imagine you can just babble on about sets, bijection, injection, infinities and so on and pretend you've said something about a subject like square circles? ~v~~
From: Jacko on 26 Nov 2006 17:04 hi just to clarify pi has a power series, all rational coefficients can be common denominatored. therefore the rational for pi has an infinite denominator. hence my term infinite rational, hence countable of order Z*Z. and this makes sense, and needs no proof by negation which would be suseptable to godelian not not red = green. and that euclidean to hyperbolic space squre circles thing ... are you guys losing it? cheers
From: Dik T. Winter on 26 Nov 2006 20:50 In article <t7pjm2pnrbh3ut0s2ll5ehvo1mi1nh26je(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Sun, 26 Nov 2006 03:32:30 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: .... > > > If you begin by assuming this you wind up by assuming that > > > and pretty soon you find yourself assuming what you were supposed to > > > demonstrate in the first place, that square circles are conceivable. > > > >Well the latter was not an assumption. There was an explicit question. > > But no demonstrated answer. You just claimed square circles are > conceivable according to a metric you can't demonstrate in the same > Euclidean terms you claim to rely on. I did. But I think you did not understand it. > > > On the > > > other hand if all you're describing are variable measures of distance > > > then you'd have to explain how you obtain those measures without an > > > underlying Euclidean metric and corresponding assumptions. You can't > > > just assume them as modern mathematikers are wont to do. > > > >You read more in the word "metric" than is in there. > > Not at all. Even in the restricted sense of "distance measure" a term > like "metric" has to be self consistent with underlying assumptions of > measure and what is measured and how. The measure I give is consistent. Any taxi-cab driver knows the metric. > >Now you require that without assumptions rather than metrics... That > >is something completely different. Let us recap the Euclidean postulates: > >(1) Two points determine a line > >(2) Any line segment can be extended in a straight line as far as desired, > > in either direction > >(3) Given any length and any point, a circle can be drawn having the length > > as radius and that point as center > >(4) All right angles are congruent > >(5) The parallel postulate > >Now, that is precisely what I did use (after translation to analytical > >geometry). > > The problem is you have a host of undefined terms for things like > points, lines, length, circles, right angles, etc.If Euclid understood > these things one way and you understand and employ them another > they're not Euclidean assumptions at all just revisionist private > definitions. To justify your claim that square circles are conceivable > you also claim a metric you don't demonstrate on the basis of > Euclidean assumptions you claim to rely on. Balderdash. > > > >In that case you should use better formulations in your questions. > > > > > > And you should use better formulations in your answers. > > > >How could I give a better formulation than stating that in the Manhattan > >measure it is possible? > > Your stating it has nothing to do with whether it's true. Later on I did show that it is true. > > I did state that explicitly, while you did not > >state that you did use Eucledian measure. > > No reason I should if the words themselves rely on the same Euclidean > assumptions you claim to rely on. They do not. > > > Except that your answer relies on non Euclidean assumptions that > > > circles are squares. > > > >No. I show that with a particular measure all circles are squares. There > >is no assumption. > > So you say but so you don't demonstrate. What you say is nothing more > than a claim. And I did show why it works. > > > If I ask why one sided quadrangles are > > > unimaginable and you reply that they aren't if you start counting from > > > four would you consider your answer responsive to the question asked? > > > >You are using a few words that you did not define, and again using some > >assumptions. Quadrangle I can understand. What is one-sided? Does it > >mean that the top side is different from the bottom side? In that case > >I can show you quadrangles on a Moebius strip that have only one side. > >On the other hand, I think you are referring to the edges. In that case, > >in topology a connected set (as a quadrangle is) has a single boundary. > > So you wind up with a four sided figure with one side? I don't see > that you're taking your words or mine seriously. How is it you get > from one side to another regardless of topology and how do you get > past the points of intersection? In topology those points are not interesting. Try to learn a bit about mathematics. > > > All you're doing is answering a question that wasn't asked in terms > > > employed by the original question. > > > >The terms employed by the original question did not refer to Euclidean > >metric. So my answer was valid. > > Terms like "circle" and "square" don't refer to Euclidean metric? Okay > then. I did define those terms in the article where I did show how it works. And those definitions are not too uncommon: A square is a rectangle where the distances between successive points are equal. and: A circle is a figure where the distances from each point on the figure to a central point are equal. What more do you want? > > > I can make up private definitions > > > just like everyone else does but that doesn't make definitions true. > > > >A definition can not be false. > > Another interesting and rather self serving claim from someone who > obviously prefers false definitions to true definitions. Indeed. Isn't it? > > It can be the case that there is nothing > >(within the current context) that satisfies the definition, but it is > >still not false. > > So is it the case that definitions cannot be self contradictory or > that self contradiction is not false? If there are contradictory requirements in the definition that does not make the definition false, it only tells us that there is nothing that satisfies the definition. Consider the following definition: a is the smallest Fermat prime larger than 65537 Is that a false definition? If so, why? And when it is shown that there are no Fermat primes larger than 65537 it suddenly becomes a false definition? > > > In fact my personal favorite private definition for distance metrics > > > is one I made up for the real number line which runs 1, 2, e, 3, pi, > > > 4, 5, . . but I don't try to pretend that when I'm trying to analyze > > > real numbers that that is a true definition. > > > >That is not enough for a definition. > > Sure it is. I just said so and you just said that definitions cannot > be false. I do not say the definition is false, I say the definition is insufficient. What is the distance between 1 and sqrt(2)? If you can not tell from your statement above, the definition is insufficient to supply a metric. > > > Besides as far as I can tell you still haven't answered my question as > > > to whether there are any curves at all with the Manhattan measure. In > > > fact I can't even find it reprinted above. > > > >Pray define "curve". > > Don't need to. For purposes of the present discussion it is enough to > understand that circles are curves and squares aren't. But in that case my answers were sufficient. There are indeed curves: the circles. They are at the same time also non-curves, but that is not a problem. (Like in topology there are sets that are both open and closed.) > If you maintain > otherwise I'd like to know how and whether there are other curved > figures besides circles in the Manhattan metric which are squares. I still do not know what you mean by "curved figures". -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Eckard Blumschein on 27 Nov 2006 03:59 On 11/24/2006 11:22 PM, Virgil wrote: >> > And one can easily imagine it as, say, as the succession of points >> > (n-1)/n in the rational open interval (0,1) which is "completed" in the >> > rational interval [0,1]. >> >> Just replacing ) by ] seems to be quite easy and natural. >> In my understanding, with really real numbers there is no difference >> between ] and ] at all. > > No one is claiming a difference between ] and ]. I apologise for my weak eyes. I meant ) and ]. > But in , say, Dedekind cuts, there is a distinguishable difference > between (0,1) and [0,1], based on whether those sets contain or do not > contain their LUBs and GLBs as members. As long as there is such difference, we are still dealing with rational numbers, not with continuity.
From: Virgil on 27 Nov 2006 04:26
In article <456AA8DE.4050606(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/24/2006 11:22 PM, Virgil wrote: > > >> > And one can easily imagine it as, say, as the succession of points > >> > (n-1)/n in the rational open interval (0,1) which is "completed" in the > >> > rational interval [0,1]. > >> > >> Just replacing ) by ] seems to be quite easy and natural. > >> In my understanding, with really real numbers there is no difference > >> between ] and ] at all. > > > > No one is claiming a difference between ] and ]. > > I apologise for my weak eyes. I meant ) and ]. > > > But in , say, Dedekind cuts, there is a distinguishable difference > > between (0,1) and [0,1], based on whether those sets contain or do not > > contain their LUBs and GLBs as members. > > As long as there is such difference, we are still dealing with rational > numbers We are dealing with sets of rationals numbers, the upper and lower Dedekind cuts. |