Cantor Confusion
in [Math]
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From: William Hughes on 17 Apr 2007 17:40 On Apr 17, 4:20 pm, mueck...(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. Piffle. Even in Wolkenmuekenheim a "point of separation" must be able to separate a path from all other paths. Since a single node cannot separate a path from all other paths a single node is not a point of separation. A set of nodes can separate a path. Therefore a set of nodes can be a "point of separation". - William Hughes
From: mueckenh on 18 Apr 2007 06:40 On 17 Apr., 23:40, William Hughes <wpihug...(a)hotmail.com> wrote: > On Apr 17, 4:20 pm, mueck...(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. > > Piffle. Even in Wolkenmuekenheim a "point of separation" must be > able to separate a path from all other paths. Piffle. A point of separation increases the number of paths which can be distinguished. By construction of the tree X nodes can create X+ 1 separated paths. > Since a single node > cannot separate > a path from all other paths > a single node is not a point of separation. > > A set of nodes can separate a path. Therefore a set of nodes > can be a "point of separation". Yes, a set of separation points can separate a path from all other paths. Not all nodes of the tree are required for this purpose. Anyhow, only countably many nodes are avaivable. Regards, WM
From: William Hughes on 18 Apr 2007 07:32 On Apr 18, 6:40 am, mueck...(a)rz.fh-augsburg.de wrote: <snip nonsense about how in Wolkenmuekenheim a separation point does not actually separate a path, but just partially separates a path, despite the fact that exactly the opposite was claimed. However, as we know consistency is not a virtue in Wolkenmuekenheim if it means that Muekenheim is incorrect.> > Yes, a set of separation points can separate a path from all other > paths. Not all nodes of the tree are required for this purpose. And even in Wolkenmuekenheim, [See if you can reply to this without mentioning "separation point" or changing the subject] there are an uncountable number of subsets of a countable set. So a countable set of nodes can separate an uncountable number of paths. - William Hughes
From: Dik T. Winter on 18 Apr 2007 10:51 In article <1176723918.670619.273750(a)q75g2000hsh.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 16 Apr., 02:19, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1176298903.609533.227...(a)d57g2000hsg.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 11 Apr., 03:17, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > Do you claim that all paths exist as separated entities? If so: > > > > > how or where do they exist? > > > > > > > > Philosophy? I would expect mathematics in this newsgroup. > > > > > > I think that numbers belong to mathematics. > > > > But the question "where do they exist" is not a mathematical question. > > If you consider the tree, it is as mathematical as if you consider a > "list". The mathematical question is: "are all paths available in the infinite tree", and the answer is: "yes". A follow-up question can be "is each path separated from all other paths", and the answer is, again, "yes". And a further question can be, "is there a level such that a particular path is separated from all other paths", and the answer to this is "no". "exist" is barely a mathematical term. > > > > This sentence is quite difficult to pars. Yes, each real number is > > > > represented by a path which is completely within the infinitely many > > > > levels of the tree. > > > > > > 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). What is a "point of separation"? Again, you are using undefined terminology. But indeed, in every finite tree with n levels there are 2^n+1 finite paths. And in the infinite tree there are countably many finite paths. But this says *nothing* about infinite paths. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 18 Apr 2007 10:56
In article <1176724396.229576.101420(a)p77g2000hsh.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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? They are all separated from each other. I do not know why you need uncountably many separation points for that. > > > Nevertheless, we can agree upon these things if we > > > clarify their meaning. Does "Every finite part of the path" mean "the > > > whole path"? Does "Every finite part of the path" include or cover > > > also that special part you just invented? > > > > And now you switch again from "finite node" to "finite part of the path". > > As "every finite part of the path" would mean to me (without further > > definition) a set of parts of paths, I would state: no, because a path > > is not a set of parts of paths but (by your own definition) a set of nodes. > > And even as a set of parts of paths, "every finite part of the path" does > > *not* contain the path itself as an element (if the path is not finite). > > What you do mean with the word "cover" escapes me. > > Every node n is in bijection with the set of nodes from the first to > n. The latter is a finite part of a path. > > If every finite node of a path means the whole path (you never > contradicted *that*), then "every finite part of the path" means "the > whole path". Wrong. A finite part of a part is (by your definition above), a *set* of nodes. Then "every finite part of the path" means (I think) the set of finite parts of the path, which is a set of sets of nodes, and so no set of nodes, and so no path. What you are stating here is similar to stating that: {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, ...} either is N or does contain N as an element. Neither is true. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |