Cantor Confusion
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From: Dik T. Winter on 25 Nov 2006 21:40 In article <1164446332.276791.309770(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1164126713.968092.237570(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > > > There are infinite paths > > > > > > in your tree, but they do not contain a node that represents (for > > > > > > instance) 1/3. So, if the nodes represent numbers (as you have said), > > > > > > > > > > Do you have a reference? > > > > > > > > Not needed. Just above you state that the nodes represent the bits 0 or > > > > 1. I have shown how you could concatenate the representation of a node > > > > with the representations of its parent nodes to get a number. > > > > > > A number consists of bits. Some numbers consist even of one bit. But > > > you must not mix up these terms. > > > A number like 1/2 consist of the bit sequence 0.1000.... That is a path > > > in my tree: > > > > As an infinite bit sequence, yes, but there is *no* node in your tree > > that represents that bit sequence. > > > > > > > > 1/3 is not in your tree. > > > > > > Of course it is, like 0.333... is in Cantor's list of decimals. > > > > No, there is *no* node in your tree that represents 1/3. Because there > > is *no* node in your tree that represents an infinite sequence. On the > > other hand, Cantor's diagonal proof is about infinite sequences. > > You do not require that one digit represents the number 1/3 in Cantor's > list. You do not require that any digit there represents an infinite > sequence. Of course not. > But you require that one digit (node) represents 1/3 in the tree? > But you require that one digit (node) in my tree represents an > infinite sequence? Of course. > Are you trolling, Dik? Not at all. > There is no node in the tree which represents 0.000... and no node > which represents 0.1 But these real numbers are represented in the > tree by paths. I do not understand. You state that the bits are the nodes. I have shown how you can extend that to the nodes representing numbers. And I also have shown how the numbers represented by the nodes all have a finite sequence of bits. There is *no* node that represents 1/3. And when you switch to paths representing numbers you have something completely different. Let's assign to a finite path the number of the node where the last edge terminates. Also in this case 1/3 is not in your paths. There are no numbers assigned to the non-terminating infinite paths. I think you wish that the edges represent the bits, not the nodes. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Nov 2006 21:46 In article <1164446451.353434.231240(a)45g2000cws.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1164126570.219569.222590(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > For geometry, it is required, yes, for arithmetic it is not. > > > > And you do not think there is an underlying "something" to which set > > theory with AC and without AC are related? > > No. So you think there is an underlyign "something" to which set theory with AC and without AC are related. What is your problem? .... > > > They have no representation which could be used for any form of > > > Cantor's list. > > > > Well, according to set theory they have such representations. > > That is one of the reasons why set theory is wrong. Nothing more than opinion. You think there are no such representations, so everything that states that there are such representations is wrong. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Nov 2006 21:50 In article <1164446537.991233.319900(a)j72g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1164126395.211430.7520(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > 3) In order to show that there is no line containing all indexes of the > > > diagonal, there must be found at least one index, which is in the > > > diagonal but not in any line. This is impossible. > > > > I do not accept (3). It effectively states: > > forall{n in N}thereis{m in N} (index m is not in line n) -> > > thereis{m in N}forall{n in N} (index m is not in line n) > > For *finite, linearly ordered* sets, the quantifiers can be exchanged. > Every line is a finite, linearly ordered set. But the sets are not the lines. The set considered is N, which is linearly ordered but not finite. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Nov 2006 22:32 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. > 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. > > 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. > 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. > > > 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). > > > 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? I did state that explicitly, while you did not state that you did use Eucledian measure. > > 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. > 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. > 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. > I can make up private definitions > just like everyone else does but that doesn't make definitions true. A definition can not be false. It can be the case that there is nothing (within the current context) that satisfies the definition, but it is still not 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. > 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". -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Ralf Bader on 26 Nov 2006 02:17
mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > >> In article <1164281187.964067.115190(a)m7g2000cwm.googlegroups.com> >> mueckenh(a)rz.fh-augsburg.de writes: ... >> > Cantor wrote: "It is remarkable that removing a countable set leaves >> > the plaine *being connected*." But it is not remarkable that removing >> > an uncountable set leaves the plaine being connected? What a foolish >> > assertion. >> >> But removing an uncountable set can leave the plane either connected or >> disconnected. What is remarkable about that? > > Nothing. In order to discuss Cantors paper one should know it. Of > course Cantor knew your trivial example and those proposed by Mr. > Bader, who is seems to be used to take trivialities for the main issue. What is this? An indication of intelligence? While you are unable to grasp anything mathematical, no matter how often it is explained to you, you are able to learn the wording of your adversaries if it comes in handy to turn it against them. Concerning Cantor, he either knew my example (which was trivial on purpose in order to make it a sure bet that he knew it) or "Cantor wanted to suggest that the removal of a countable set leaves the plane connected while the removal of an uncountable set does not.", as you wrote in a previous posting. It is well known that words can always change their meaning in your babble, but that it goes that fast is amazing. Can you add up 2+2 without making an imbecile mess out of it? Is what you do here a weird psychological experiment to find out how much idiotic and self-contradictory babble is necessary to drive others crazy? This is implausible if one thinks of the amount of typing work gone into your scrappy papers and your stamina in delivering foolish newsgroup postings. On the other hand it is also implausible that a single person should be as stupid as you present yourself. So we are faced with a real problem. R.B. |