Cantor Confusion
in [Math]
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From: mueckenh on 17 Apr 2007 11:14 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. Regards, WM
From: William Hughes on 17 Apr 2007 12:11 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. - William Hughes
From: Virgil on 17 Apr 2007 14:34 In article <1176822676.172190.96100(a)y80g2000hsf.googlegroups.com>, mueckenh(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? > > > > 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. 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. > > > 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. > But there are only countably many separations and > countably many points of separation. Then there are none of either. A "point of separation" separating one path in a CIBT from all others is never a single node, it is an at least infinite subset of the set of nodes of that path, which we might as well identify with the path itself. Note that in a finite tree, it is only leaf nodes which allows separation of the paths containing them from all others. So either WM is claiming CIBT's have leaf nodes, or he must allow that only infinite sets of nodes allow separation of one path from all others in CIBTs.
From: Virgil on 17 Apr 2007 14:39 In article <1176822855.764056.103810(a)y80g2000hsf.googlegroups.com>, mueckenh(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. And as each path, as an infinite set of its own set of nodes, creates a separate separation, there are as many such separations as paths, which is still uncountable.
From: mueckenh on 17 Apr 2007 16:20
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. Regards, WM |