Cantor Confusion
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From: Dik T. Winter on 27 Nov 2006 21:01 In article <17mmm21mhdro54dilk6n9qk3r7chagfoqk(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Mon, 27 Nov 2006 01:50:38 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: .... > >I did. But I think you did not understand it. > > Of course I didn't understand in the Euclidean terms you claim to rely > on. The more important question is whether anyone can understand it. In that case you should first study the Euclidean terms. > > > 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. > > Well I rather doubt that. Maybe Manhattan taxicab drivers know the > metric. And maybe that's why they're taxicab drivers. Oh, well, if they understand it and you do not... > > > 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. > > They're kind of interesting when you try to get around them without > specifying how the trick is done. But then maybe we should consult a > taxicab driver. Except I have no desire to insult taxicab drivers. You do not understand topology. That is clear. > >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? > > How about a little less casuistry for a change? I have no idea where > you get these nonsensical definitions for squares and circles. "A > square is a rectangle"? So what is a rectangle? Are the sides of > rectangles straight lines? Are circles composed of straight lines? I gave precisely those definitions in the original article. But you just ignored them. I even did define straight lines. > > 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? > > Sure. Why not? Well, mathematicians prefer to have definitions to remain true, whether it is later shown that there are no things that satisfy the definition or not. That you prefer to use definitions that suddenly can become false is just your opinion. > > > 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. > > Then you say a lot of things which are 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. > > Only in your dreams. I can most definitely show the exact distance > between 1 and the square root of two using rac construction. You are completely outfield. What *is* rac construction? > We're > still waiting for you to show us this magic Manhattan metric using > anything but your daydreams. I defined it. What more can I do? > > > 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. > > But in that case your answers are most definitely sufficient just not > demonstrably true. Did you look at what I did define? I think not. > > 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.) > > Yes well I daresay so: "There are indeed curves but they are also non > curves but that is not a problem" because modern mathematikers are too > lazy or stupid to consider the truth of what they're talking about. You have completely lost me. > > > 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". > > Then learn some mathematics. Maybe your true calling is indeed driving > taxicabs. I do not think so. I will continue doing mathematics (and trying to understand it), but it would be better if you did try to understand it, which you obviously do not. I stop this discussion. Discussing with someone who loaths mathematics about mathematics does not make sense. -- 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 28 Nov 2006 02:42 mueckenh(a)rz.fh-augsburg.de wrote: > > > Ralf Bader schrieb: > >> 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. >> >> Concerning Cantor, he either knew my example (which was trivial on >> purpose in order to make it a sure bet that he knew it) > > Your example is in fact trivial: "E.g. the points inside a circle or > rectangle, and I'm pretty sure that Cantor was aware of this." But it > is even more trivial to see that Cantor did not consider it at all > because he required the removed set do be dense in the whole plane. I > wrote that already in the text wisely snipped by you. Of course you > cannot agree to having been mistaken. O perhaps you really don't > recognize it? Mein lieber Scholli, that Cantor did not consider such examples at *that* place is already a consequence of the fact that he dealt there with countable sets only. If Cantor makes another assumption which I wasn't aware of and which also excludes those rectangles then this does not add anything to the already established matter of the fact - that he does not consider those examples at that place. Learn some logic. Learn to make clear statements instead of arbitrarily smuggling in additional assumptions. Moreover, you should try to find out where in Cantor's proof the density of M is actually used. Come back if you have found the answer. Maybe this answer has something to do with why I neglected this density. Hint: Not everything Cantor wrote is just perfect... > A merciful hint: To concentrate the words babble, imbecile mess, > idiotic, foolish, stupid within few lines is not very convincing. And I used those words because they were necessary for an appropriate statement of fact, not to convince anybody of anything. Moreover, using those words is a concession to youu because it allows you to comment on my language instead of the issue at hand where you have nothing sensible to say. > slandering in a math newsgroup is not very impressive if the slanderer > mistakes the set of rational numbers for the set of algebraic numbers. In the context of your "simplified" proof this difference does not matter. In fact, if M is an arbitrary subset of R and n>1, then the subset S of R^n consisting of all points not all of whose coordinate values are from M is path-connected (proof: If M=R then S is empty, so it is connected. Otherwise, let t be in R\M. Then the point p=(t,...,t) is in S, and it suffices to construct a path from an arbitrary point q=(q_i) of S to p. There is some q_r in R\M. Homotope all coordinate values of q to t, except the r.th which is kept constant. This is a path s in S, ending in a point q'. Join another path to s which is obtained by homotoping q'_r to t, keeping all other coordinates constant. This is also a path in S which ends in p.) > BTW: Have you meanwhile learnt to recognize the Greek letter xi and to > distinguish it from the Greek letter zeta? I've learnt what a pettifogger is. ....and not everything what I write is perfect. Concerning my errors, there is proof that I admit them. Only you, Mr. Grömaz Mückenpein, don't commit any errors. In your rendering of Cantor's proof in §A2 of your shitty paper, you write " We consider a finite set of them {N1, N2 ,..., Nk}". When I read this, I wondered why you don't just take the two endpoints. Reading Cantor's original proof, the reason for introducing those points becomes clear: They are needed to stay within the region A. But you dropped A, and therefore those points N_i are an unnecessary complication in your "simplified" proof, which btw isn't a simplified proof at all because one expects from a simplified proof that it proves (at least) as much as the original which isn't the case with our proof. The deeper one digs into your pile of junk, the more absurdities appear. Your understanding of Cantor's obviously is poor, and not only at this point. In §2 of your paper "A severe inconsistency of transfinite set theory" you dropped the fact that the collection of intervals {\Delta_\perplexon} resulted from the analysis of a perfect set S, as explained by Cantor on the previous page, and therefore isn't just arbitrary. Contrary to certain other assumptions, this one is decisive, and the consequences you draw from your misunderstanding are simply wrong. Concerning wisely snipped out text, you wrote in Message-ID: <1164208580.112906.57510(a)b28g2000cwb.googlegroups.com> "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 an example he chose the algebraic numbers (obviously in order to distinguish them from the transcendental numbers)." Instead of explaining on which evidence you base *your* assertion that "Cantor wanted to suggest that ...the removal of an uncountable set does not" you simply cut this question off and ramble in a hodge-podge of trivialities and distortions. On top of all, even if only dense subsets are considered, the rectangle union all points with only rational (or algebraic) coordinates is a dense uncountable set. Still more easy: the complement of a point in R^n is a dense uncountable subset whose removal leaves a remainder which is connected. But it is already an unwarranted concession that I try to make it plausible that Cantor did not want to suggest what you impute to him. It is *your* business to substantiate your assertions. So do this or else *you* are a slanderer - and the only one, btw. Concerning the difference between rational and algebraic numbers, which you accuse me in blindfold stupidity to be unaware of, there is, I think, a homeomorphism r: R->R carrying the set of rational numbers onto the set of algebraic numbers. So the difference between rational and algebraic numbers is *topologically* inexistent. R.B.
From: Bob Kolker on 28 Nov 2006 07:35 Eckard Blumschein wrote: > > > Parochial terms tend to be somewhat misleading. > I recall that I wondered some 50 years ago why the term vector denotes > something geometrical in physics, while it also means a set of numbers Forces have magnitude and direction. Ditto momentum. > in mathematics. I already understood: Geometry is continuous while > nubers are discrete. A vector can be represented by an n-tuple of numbers, not just any old unordered set of numbers. Ignorant putz. Bob Kolker
From: Lester Zick on 28 Nov 2006 14:31 On Tue, 28 Nov 2006 02:01:13 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <17mmm21mhdro54dilk6n9qk3r7chagfoqk(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > On Mon, 27 Nov 2006 01:50:38 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: >... > > >I did. But I think you did not understand it. > > > > Of course I didn't understand in the Euclidean terms you claim to rely > > on. The more important question is whether anyone can understand it. > >In that case you should first study the Euclidean terms. Why? You have yet to show any connection. > > > > 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. > > > > Well I rather doubt that. Maybe Manhattan taxicab drivers know the > > metric. And maybe that's why they're taxicab drivers. > >Oh, well, if they understand it and you do not... Then they probably deserve to be taxicab drivers. > > > > 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. > > > > They're kind of interesting when you try to get around them without > > specifying how the trick is done. But then maybe we should consult a > > taxicab driver. Except I have no desire to insult taxicab drivers. > >You do not understand topology. That is clear. However I understand people who try to get around points but who can't say how the trick is done. They're called mathemagicians and empirics. > > >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? > > > > How about a little less casuistry for a change? I have no idea where > > you get these nonsensical definitions for squares and circles. "A > > square is a rectangle"? So what is a rectangle? Are the sides of > > rectangles straight lines? Are circles composed of straight lines? > >I gave precisely those definitions in the original article. But you >just ignored them. I even did define straight lines. Yeah I remember. What you didn't say however was how it is that circles are composed of straight lines. That's the issue and not how and whether taxicab drivers in Manhattan think differently. > > > 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? > > > > Sure. Why not? > >Well, mathematicians prefer to have definitions to remain true, whether >it is later shown that there are no things that satisfy the definition >or not. So they just assume what they say is true whether it turns out to be or not? That's real bright. Maybe you really are a taxicab driver. > That you prefer to use definitions that suddenly can become >false is just your opinion. And of course it's not "just your opinion" that definitions cannot be problematic. > > > > 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. > > > > Then you say a lot of things which are 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. > > > > Only in your dreams. I can most definitely show the exact distance > > between 1 and the square root of two using rac construction. > >You are completely outfield. What *is* rac construction? Learn some mathematics for a change. > > We're > > still waiting for you to show us this magic Manhattan metric using > > anything but your daydreams. > >I defined it. What more can I do? Explain how it is that circles are composed of straight lines. > > > > 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. > > > > But in that case your answers are most definitely sufficient just not > > demonstrably true. > >Did you look at what I did define? I think not. I looked at what you didn't define: how circles are composed of straight lines. You're an idiot. > > > 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.) > > > > Yes well I daresay so: "There are indeed curves but they are also non > > curves but that is not a problem" because modern mathematikers are too > > lazy or stupid to consider the truth of what they're talking about. > >You have completely lost me. You completely lost yourself a long time ago. > > > > 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". > > > > Then learn some mathematics. Maybe your true calling is indeed driving > > taxicabs. > >I do not think so. Who cares what you think. It's what you can demonstrate that matters. > I will continue doing mathematics (and trying to >understand it), but it would be better if you did try to understand it, >which you obviously do not. Of course I don't understand mathematics. I just understand that circles are not composed of straight lines. > I stop this discussion. What discussion? > Discussing with >someone who loaths mathematics about mathematics does not make sense. Good. Then maybe you should go back to growing tulips and let the truth fairy do your mathematical thinking for you. ~v~~
From: mueckenh on 29 Nov 2006 11:13
MoeBlee schrieb: > > I think if one swells to explosion about his knowledge of set theory, > > he should at least know the very foundation. But I know, that you do > > not even understand the simple texts of Fraenkel et al. > > No, YOU radically MISunderstand what Fraenkel, Bar-Hillel, and Levy > wrote. How can you judge about that without the slightest idea of what they wrote? Only for the lurkers: Fraenkel et al. write: "Platonistic point of view is to look at the universe of all sets not as a fixed entity but as an entity capable of "growing", i.e., we are able to "produce" bigger and bigger sets." So a set (like the set of all sets) is not a fixed entity. >Meanwhile, you are ignorant of even such basic set theory as > proving the existence of an identity function. I think of a correct proof, not of blowing hot air. > > As to cardinality, See Levy's 'Basic Set Theory' for a brief discussion > of Scott's method. > > > > > PS: What about the binary tree in ZFC? Any results? Any idea why it > > > > can't be done? > > > > > > I already mentioned what I would require to discuss your tree argument. > > > If you want me to talk about your tree, then please answer my previous > > > posts from about a week ago on the subject of even discussing your > > > argument. I put it to you very clearly just what you need to tell me > > > before we proceed. If you won't do so, then I'm not inclined to waste > > > my time on your argument. > > > > What I can tell you is the following: > > > > The binary tree > > > > Consider a binary tree which has (no finite paths but only) infinite > > paths representing the real numbers between 0 and 1 as binary strings. > > The edges (like a, b, and c below) connect the nodes, i.e., the binary > > digits 0 or 1. > > > > 0. > > /a \ > > 0 1 > > /b \c / \ > > 0 1 0 1 > > .......................... > > > > The set of edges is countable, because we can enumerate them. Now we > > set up a relation between paths and edges. Relate edge a to all paths > > which begin with 0.0. Relate edge b to all paths which begin with 0.00 > > and relate edge c to all paths which begin with 0.01. Half of edge a is > > inherited by all paths which begin with 0.00, the other half of edge a > > is inherited by all paths which begin with 0.01. Continuing in this > > manner in infinity, we see by the infinite recursion > > > > f(n+1) = 1 + f(n)/2 > > > > with f(1) = 1 that for n --> oo > > > > 1 + 1/2 + 1/ 4 + ... = 2 > > > > edges are related to every single infinite path which are not related > > to any other path. (By the way, the recursion would yield the limit > > value 2 for any starting value f(1).) The load of 2 edges is only > > related to infinite paths because any finite segment of a path with n > > edges will carry a load of > > > > (1 - 1/2^n)/(1 - 1/2) < 2 > > > > edges. The set of paths is uncountable, but as we have seen, it > > contains less elements than the set of edges. Cantor's diagonal > > argument does not apply in this case, because the tree contains all > > binary representations of real numbers within [0, 1], some of them even > > twice, like 1.000... and 0.111... . Therefore we have a contradiction: > > > > |IR| > |IN| > > || || > > |{paths}| =< |{edges}| > > > > If you don't understand this simple and clear exposition, then there is > > no hope that you will be able to think any further. > > > > I really don't know what further information could be required. > > The information I require about the very basis of this discussion was > specified in my previous posts - the ones I wrote a couple of week ago > and which I mentioned yet again, but which you yet again ignore. I took > the effort to compose those questions for you at that time. If you want > me to consider your argument, then please respond to those posts. I'm > not inclined to recompose my questions for you again after I already > carefully composed them. Which posts? When posted? By copy and paste you should be able to repeat those questions. I saw only the question whether I believed that a proof in ZFC was possible. Should this question decide whether such a proof is really possible? Regards, WM |