Cantor Confusion
in [Math]
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From: Virgil on 16 Apr 2007 14:33 In article <1176726344.265077.211810(a)n59g2000hsh.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 16 Apr., 02:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1176300643.280458.120...(a)e65g2000hsc.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > > > On 11 Apr., 03:29, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > > > > Without nature (or reality, as we say today) there would not be any > > > > > natural number. > > > > > > > > And, so what? If I now define that the natural numbers are the > > > > ordinal > > > > numbers, there is nothing wrong with that. It is just contrary to > > > > common > > > > nomenclature. So, no, finiteness of the natural numbers is *not* > > > > because > > > > of nature, but it is because of definition. > > > > > > You could define a triangle with four corners. It would be as > > > meaningful and as possible. > > > > Of course. Definitions are not wrong, in principle. But by the common > > definitions natural numbers are limited to be finite. Just a case of > > defining things. There is nothing in nature, or reality, that (at least > > for mathematics) *mandates* that they should be finite. It is their > > mathematical definition that makes them finite. > > No. It is that they are obtained from natural sets of distinguishable > elements. But there are no infinite sets in nature or reality. "Nature" and "reality" may suggest what is developed in mathematics, but in no way limit mathematical development to only what they suggest.
From: Virgil on 16 Apr 2007 14:40 In article <1176731602.159201.47070(a)o5g2000hsb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 16 Apr., 14:27, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > On Apr 16, 7:53 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > > > > 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". > > > > So what? > > > > i finite parts of the path have property X > > > > ii the whole path is the union of the finite parts of the path > > > > This does not imply that > > > > iii the whole path has property X. > > Look for the answer of Dik T. Winter: "the whole path is not the union > of the finite parts of the path" > > Guess why? Because those 'finite parts' are not merely sets? Unions of things which are not mere sets may not exist at all, and even if they exist as unions of sets they need not have the properties of their constituent parts. WM habitually elides a lot of mathematically necessary issues to jump to his unjustifiable conclusions.
From: mueckenh on 16 Apr 2007 16:17 On 16 Apr., 16:31, "William Hughes" <wpihug...(a)hotmail.com> wrote: > On Apr 16, 9:53 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > > On 16 Apr., 14:27, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > On Apr 16, 7:53 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > 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". > > > > So what? > > > > i finite parts of the path have property X > > > > ii the whole path is the union of the finite parts of the path > > > > This does not imply that > > > > iii the whole path has property X. > > > Look for the answer of Dik T. Winter: "the whole path is not the union > > of the finite parts of the path" > > > Guess why? > > > Regards, WM > > Only someone with no ethics at all would use quotation marks > to indicate a paraphase. Dik Winter did not say > > the whole path is not the union > of the finite parts of the path > > Nothing that he did say contradicts this. The "special > part" is a figment of your imagination. > > - William Hughes- Zitierten Text ausblenden - > > - Zitierten Text anzeigen - He said: "Every finite part of the path does *not* mean all parts of the path." All parts = is the whole path. Regards, WM
From: William Hughes on 16 Apr 2007 16:51 On Apr 16, 4:17 pm, mueck...(a)rz.fh-augsburg.de wrote: > On 16 Apr., 16:31, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > On Apr 16, 9:53 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > On 16 Apr., 14:27, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > On Apr 16, 7:53 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > 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". > > > > > So what? > > > > > i finite parts of the path have property X > > > > > ii the whole path is the union of the finite parts of the path > > > > > This does not imply that > > > > > iii the whole path has property X. > > > > Look for the answer of Dik T. Winter: "the whole path is not the union > > > of the finite parts of the path" > > > > Guess why? > > > > Regards, WM > > > Only someone with no ethics at all would use quotation marks > > to indicate a paraphase. Dik Winter did not say > > > the whole path is not the union > > of the finite parts of the path > > > Nothing that he did say contradicts this. The "special > > part" is a figment of your imagination. > > > - William Hughes- Zitierten Text ausblenden - > > > - Zitierten Text anzeigen - > > He said: > "Every finite part of the path does *not* mean all parts of the > path." > > All parts = is the whole path. > > Regards, WM In other words what you put in quotes is not what he said. You are of course going to issue an abject appology. - William Hughes
From: mueckenh on 17 Apr 2007 11:11
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 or sufficient to > separate one path from all others in a CIBT. Wrong. It presumes correctly that one node is necessary to create a separation. > > 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. > 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. But there are only countably many separations and countably many points of separation. Regards, WM |