Cantor Confusion
in [Math]
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From: mueckenh on 17 Apr 2007 16:24 On 17 Apr., 20:34, Virgil <vir...(a)comcast.net> wrote: > In article <1176822676.172190.96...(a)y80g2000hsf.googlegroups.com>, > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > On 16 Apr., 20:23, Virgil <vir...(a)comcast.net> wrote: > > > In article <1176723918.670619.273...(a)q75g2000hsh.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > > > But not all real numbers coexist in the tree? Only a countable number > > > > > > of them is admitted simultaneously? > > > > > > Why? > > > > > There cannot be more separated paths in the whole tree than are points > > > > of separation in the whole tree, (unless there is more than one > > > > separation per point of separation which, however, can be excluded by > > > > the construction of the tree). > > > > That presumes, falsely, that one node is necessary and sufficient to > > > separate one path from all others in a CIBT. > > > Wrong. It presumes correctly that one node is necessary to create a > > separation. > > Of what from what? > Of two (bunches of) paths which up to then run together. > > > It takes infinitely many nodes in a CIBT to simultaneously separate any > > > one path from all other paths. > > > There are infinitely many nodes in the tree. > > There are even infinitely many nodes in any one path. Yes. > > Any infinite subset of the nodes in one path is sufficient to separate > that path from all others, but no finite set of nodes suffices. Yes. > > > > > > All any one node can do is to separate an > > > uncountable set of paths fro another uncountable set of paths. > > > It is not necessary to separate this or that fiction. What we have is: > > In the CIBT there are uncountably many separated paths existing - > > separated from one another. Hence there are uncountably many > > separations. > > As it takes an infinite set of nodes to achieve separation of one path > from all others, those separators are are infinite sets of nodes, not > single nodes, and are as many as the set of all subsets of the set of > all nodes. Every separation makes two (bunches of) paths from one (bunch of) path. The total number of all separated (bunches of) paths in the whole tree is exactly the number of nodes of the whole tree. Regards, WM
From: mueckenh on 17 Apr 2007 16:26 On 17 Apr., 20:39, Virgil <vir...(a)comcast.net> wrote: > In article <1176822855.764056.103...(a)y80g2000hsf.googlegroups.com>, > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > On 16 Apr., 20:29, Virgil <vir...(a)comcast.net> wrote: > > > In article <1176724396.229576.101...(a)p77g2000hsh.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > On 16 Apr., 02:43, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > In article <1176300393.408149.165...(a)w1g2000hsg.googlegroups.com> > > > > > mueck...(a)rz.fh-augsburg.de writes: > > > > > > > If all paths exist simultaneously, then there must exist uncountably > > > > > > many in the infinite tree. > > > > > > But there are. > > > > > Without the chance that uncountably many are separated in the whole > > > > tree? > > > > Uncountably many paths are separated from another uncountably many at > > > every node in that infinite tree. To separate any one path from all > > > others takes a set of countably many nodes. > > > But there are in the CIBT uncountably many paths, each one of them > > being separated from all others. Alas, there are only countably many > > separations. > > Actually, there re uncountably many separators for the separation of any > one fixed path from all others as there are uncountably many infinite > subsets of the set of nodes of that path, any one of which will serve. In the whole tree there are only countably many separators (= nodes). Regards, WM
From: Virgil on 17 Apr 2007 16:58 In article <1176841201.210091.283260(a)n76g2000hsh.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 17 Apr., 18:11, William Hughes <wpihug...(a)hotmail.com> wrote: > > On Apr 17, 11:11 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > there are only countably many separations and > > > countably many points of separation. > > > > No, each "point of separation" is a set of nodes. There are > > uncountable many sets of nodes. > > Each "point of separation" is a single node. There are countably many > nodes. Such a "point of separation" in a CIBT can only separate one uncountable set of paths from another uncountable set of paths. To separate one path in a CIBT from all others takes a set of infinitely many nodes. So which of those two sorts of "separation" is WM talking about?
From: Virgil on 17 Apr 2007 17:09 In article <1176841494.195447.215230(a)e65g2000hsc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 17 Apr., 20:34, Virgil <vir...(a)comcast.net> wrote: > > In article <1176822676.172190.96...(a)y80g2000hsf.googlegroups.com>, > > > > > > > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 16 Apr., 20:23, Virgil <vir...(a)comcast.net> wrote: > > > > In article <1176723918.670619.273...(a)q75g2000hsh.googlegroups.com>, > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > > > > But not all real numbers coexist in the tree? Only a countable > > > > > > > number > > > > > > > of them is admitted simultaneously? > > > > > > > > Why? > > > > > > > There cannot be more separated paths in the whole tree than are > > > > > points > > > > > of separation in the whole tree, (unless there is more than one > > > > > separation per point of separation which, however, can be excluded by > > > > > the construction of the tree). > > > > > > That presumes, falsely, that one node is necessary and sufficient to > > > > separate one path from all others in a CIBT. > > > > > Wrong. It presumes correctly that one node is necessary to create a > > > separation. > > > > Of what from what? > > > Of two (bunches of) paths which up to then run together. One node only separates two uncountable "bunches" of paths from each other. > > > > > > It takes infinitely many nodes in a CIBT to simultaneously separate any > > > > one path from all other paths. > > > > > There are infinitely many nodes in the tree. > > > > There are even infinitely many nodes in any one path. > > Yes. > > > > Any infinite subset of the nodes in one path is sufficient to separate > > that path from all others, but no finite set of nodes suffices. > > Yes. > > > > > > > > > > All any one node can do is to separate an > > > > uncountable set of paths from another uncountable set of paths. > > > > > It is not necessary to separate this or that fiction. What we have is: > > > In the CIBT there are uncountably many separated paths existing - > > > separated from one another. Hence there are uncountably many > > > separations. > > > > As it takes an infinite set of nodes to achieve separation of one path > > from all others, those separators are are infinite sets of nodes, not > > single nodes, and are as many as the set of all subsets of the set of > > all nodes. > > Every separation makes two (bunches of) paths from one (bunch of) > path. The total number of all separated (bunches of) paths in the > whole tree is exactly the number of nodes of the whole tree. That assumes, contrary to fact, that each subset of the set of paths can be separated from its complimentary set of paths by a single node. That is clearly not the case. E.g., the set of all paths branching left at some level other than the root the root cannot be separated from its complimentary set of paths by any single node. And there are uncountably many other examples which also do not fit WM's assumption.
From: Virgil on 17 Apr 2007 17:14
In article <1176841576.212010.171970(a)q75g2000hsh.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 17 Apr., 20:39, Virgil <vir...(a)comcast.net> wrote: > > In article <1176822855.764056.103...(a)y80g2000hsf.googlegroups.com>, > > > > > > > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 16 Apr., 20:29, Virgil <vir...(a)comcast.net> wrote: > > > > In article <1176724396.229576.101...(a)p77g2000hsh.googlegroups.com>, > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > > On 16 Apr., 02:43, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > > In article <1176300393.408149.165...(a)w1g2000hsg.googlegroups.com> > > > > > > mueck...(a)rz.fh-augsburg.de writes: > > > > > > > > > If all paths exist simultaneously, then there must exist > > > > > > > uncountably > > > > > > > many in the infinite tree. > > > > > > > > But there are. > > > > > > > Without the chance that uncountably many are separated in the whole > > > > > tree? > > > > > > Uncountably many paths are separated from another uncountably many at > > > > every node in that infinite tree. To separate any one path from all > > > > others takes a set of countably many nodes. > > > > > But there are in the CIBT uncountably many paths, each one of them > > > being separated from all others. Alas, there are only countably many > > > separations. > > > > Actually, there re uncountably many separators for the separation of any > > one fixed path from all others as there are uncountably many infinite > > subsets of the set of nodes of that path, any one of which will serve. > > In the whole tree there are only countably many separators (= nodes). As no node separates any path from all other paths, single nodes are not separators in that sense. And there are lots of partitionings of the set of all paths into two disjoint sets of paths which are impossible to achieve by any single node, so there are lots more "separations" than nodes. WM's false claim that somehow analysis of "separations" will establish countability, will continue to fail, no matter how often he resorts to it. |