Cantor Confusion
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From: Virgil on 18 Jan 2007 15:31 In article <1169111252.322725.263330(a)a75g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > T1, the union of all finite trees, contains only finite paths. > > It would be nice if you could show the existence of at least one finite > path in this union. Can you? Where does a path end? With its last (terminal or leaf) node, which in a finite tree every path has. > > > If you define a tree to be a set of paths, then > > two different trees can have the same nodes. > > My trees are defined to be a set of nodes and of edges. That is an improvement. Previously, WM's trees only had nodes, > Every > uninterrupted sequence of edges is called a path. Therefore my finite > trees and T1 and T2 are trees. Non sequitur. One can have a set of such paths that does not form a tree. > > But I do not define a tree to be a set of paths ! But you do not define it the way mathematicians do, either. http://mathworld.wolfram.com/BinaryTree.html
From: Virgil on 18 Jan 2007 15:41 In article <1169111380.377993.67320(a)l53g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > As I have mentioned in my last post: You have not given a definition of > > a union of all finite trees. > > > The union of two finite trees T(m) and T(n) with m and n levels, > respectively, where m < n, is the tree with n levels. This definition > unites sets of nodes (and sets of edges, respectively) and it is valid > for Cut Trees (CT) as well as for trees of type Weeping Willow (WWT). Then it is not just a "union of trees" but a nesting of 2 trees with one tree being a subtree of the other. > > The union of all finite trees is the union of all trees with n levels > where n is a natural number: > UT = T(1) U T(2) U T(3) U ... > Since according to WM's "definition" one is unioning sets of nodes and sets of edges and sets of paths to form the "union" of two trees, one must note that the union of all those sets of paths is like the union of all the finite initial sets of naturals in that it only contains finite objects as members (that it is an infinite set of paths does not make it a set of inifinite paths). So WM's "union" still does not contain any infinite paths, and certainly not all infinite paths as he wrongly claims.
From: Virgil on 18 Jan 2007 15:43 In article <1169111577.984314.5300(a)v45g2000cwv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Is your "big tree" the union of all subtrees? Or is it more (or less)? WM's is less, because he claims his is has only countably many paths whereas the full tree has uncountably many.
From: Virgil on 18 Jan 2007 15:45 In article <1169113252.893339.122810(a)s34g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I will not further reply to this silly arguing. > > Rgards, WM We, on the other hand WILL contintue to rely to WM's silly arguing.
From: Virgil on 18 Jan 2007 15:54
In article <1169113698.756551.125450(a)q2g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Since every path in every one of the incomplete infinite binary trees in > > the union is 'eventually constant', every path in the union of all those > > trees will also be eventually constant. > > Wrong. > Every segment {1,2,3,...,n} is finite, but the union of all segments is > not finite. That union is not a member of itself. > Every finite segment of the path 0.010101... has only finitely many > switches 0 to 1 and 1 to 0, but the union of al finite segments has > infinitely many switches. The union of any collection of only finite segments contains only finite segments. You are again conflating the union as a set with its members. > > > > > Thus WM's argument that the union of finite binary trees generates a > > complete infinite binary tree is falsified. In my union of infinite incomplete binary trees in which every path is eventually constant, this union includes every node and every edge of the complete tree, but does not contain any path which does not become eventually constant. This sort of incomplete infinite tree is what WM's union actually produces. It contains only countably many paths, but the complete tree contains uncountably many. |