Cantor Confusion
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From: David Marcus on 23 Jan 2007 15:45 imaginatorium(a)despammed.com wrote: > David Marcus wrote: > > > As for Cantor, his great idea was that it made sense to compare sets by > > whether you could biject or inject them. Before him, people thought that > > this idea didn't work. > > I don't understand what you mean by this - I thought that "before > Cantor", people just assumed that the only size "beyond any finite > size" would be "infinite". You could always line up two infinite sets > side by side (i.e. in a bijection), and start counting, and neither > would ever end, so you couldn't say that either was bigger than the > other. > > Do you mean that they thought something else entirely? No, that's not what I meant. For things like circles and line segments, my impression is they thought that nothing useful could come from considering bijections. You can set up a bijection between the points of two circles of different size, but this seems of no use since we know that one circle is bigger than another. It was Cantor who realized that considering such bijections is actually very useful. That's all I meant by "didn't work", i.e., didn't produce useful results or mathematics. Although, I'm not sure if people assumed you could line up the naturals and the reals side by side. I wonder if anyone even considered doing such a thing. -- David Marcus
From: Virgil on 23 Jan 2007 15:54 In article <1169551459.707075.42900(a)v45g2000cwv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1169473008.561623.314970(a)v45g2000cwv.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > > > > The minimal set of paths consistent with a tree being infinite is > > > > countable but the maximal one is not. > > > > > > The minimal set is the maximal set. > > > Specifically the set of all paths which are "eventually constant" is a > > proper subset of the set of all possible paths, but it still uses every > > node and every edge. > > Correct. This shows that there are no other paths. It is the core of my > contradiction of set theory. Then the infinitely many paths that from some node onwards alternate left branch then right branch do not exist? And none of the infinitely many other infinitely alternating patterns exist? > Everything else claimed by you is belief in ghost paths (vibrations of > vacuum? small black holes in a green tree?), but has nothing at all to > do with mathematics. There are all sorts of proper fractions with odd denominators which have binary expansions corresponding to paths that WM says do not exist. It would seem that these "vibrations in a vacuum" are taking place inside WM's head and his small black holes are eating away what little logic he has left. > > > > > > > > > As the distinction between the set of paths being countable and > > > > uncountable is what WM is interested in, those sets of paths must be > > > > taken into consideration. > > > > > > Both sets are the same countable set. > > > > As per my example above, the set of "eventually constant" paths is > > countable (as can be easily shown) but the set of all paths is not (as > > Cantor proved with his original diagonal argument). > > How did he prove that? See above! > Did he really advance to infinity? Did he really > surpass the nodes of the union of trees AND the paths of the union of > trees? Did anybody, two weeks ago, believer, that there are different > sets of paths in the same tree, coming and going like sweet little > birds, perhaps even whistling and twittering? WM seems to have left the rational world of mathematics to indulge himself in poetic musings. > > No, he said that for a_nn we can put b_nn and claimed that this is > valid for every n. I claim that a path touching a node will also touch > the next one and that all possible combinations are realized in an > infinite tree with all nodes, > > Why do you believe that in Cantor's matrix there is only one fixed kind > of diagonals? Why does WM keep making these wild and totally unfounded claims that we believe things we do not believe? > According to your arguing, there are many choices. Which WM proves by always making wrong ones.
From: Virgil on 23 Jan 2007 16:01 In article <1169552015.252115.167750(a)j27g2000cwj.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > The union of any finite number of finite segments is a finite segment, > > but the union of infinitely many finite segments is not a finite > > segment, so that WM's argument fails. > > Sorry, that IS my argument. The union of all finite trees is not a > finite tree. Therefore it has no finite paths. The union, in any mathematical sense of what a union is, of two or more distinct trees is not even a tree, but it includes, in a way, all the paths of all the separate trees which it conjoins. > But also: All finite trees contain only finite paths. Therefore the > union of all finite trees contains no infinite paths. True, but not in the sense that WM is asserting it. There is a way of conjoining trees to form a tree, but it is not by set theoretic unions, as WM claims. In such a conjunction of all finite binary trees, the result is an infinite binary tree with uncountably many paths. > > Result: There is no infinite union of elements of any kind. Wrong, as usual. It is just that such a /union/ is not itself a tree.
From: Virgil on 23 Jan 2007 16:04 In article <1169552240.187770.24670(a)k78g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1169489693.112797.92660(a)a75g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > WM is not competent to pass judgement on mathematicians. > > > > > > That could be decided by mathematicians only. > > > > Right! And WM is not one of them. > > That could be decided by mathematicians only. Several mathematicians have spoken on that issue, and they have uniformly judged WM not to be a mathematician. So my own opinion is widely shared by those whose reputations here as mathematicians are beyond dispute.
From: Virgil on 23 Jan 2007 16:09
In article <1169553167.673110.130610(a)l53g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > Virgil wrote: > > > > So you obviously use a different notion > > of tree. > > He does not use any notion but the fact, as he believes, that the set > of all paths in a complete tree must be uncountable. As he (and I, and Cantor, and others) can prove that the set of paths in a complete infinite binary tree are unlistable (= uncountable), his belief is justified true belief. > The nodes of a > complete tree (complete concerning nodes and edges) are already > completely occupied by paths which are in the union of all finite trees > (or trees of type weeping willow). Claimed without proof. And WM's notion of "union" is either not a union or is not a tree. |