In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 13 Mrz., 19:19, fom <fomJ...@nyms.net> wrote: > > On 3/13/2013 12:33 PM, WM wrote: > > > > > > > > > > > > > On 13 Mrz., 17:59, William Hughes <wpihug...@gmail.com> wrote: > > >> On Mar 13, 5:37 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > >>> On 13 Mrz., 13:19, William Hughes <wpihug...@gmail.com> wrote: > > > > >> <snip> > > > > >>>> If you wish to contest this, use my words not > > >>>> yours (e.g. I have never said "The list contains more > > >>>> numbers than fit into a single line", I have said > > >>>> "There is no line in the list which contains every > > >>>> number in the list".) > > > > >>> Correct. The list has more numbers than a single line has. Since every > > >>> number that is in the list, must be in at least one line, this implies > > >>> that the numbers are in more than one line. > > > > >> To be precise, a set of lines, say K, that contains all the numbers > > >> contains at least two lines. > > > > > In actual infinity, this is not avoidable. > > > We note: At least two lines belong to the set that contains all > > > numbers. We call these lines necessary lines. > > > So the set of necessary lines is not empty. > > > > >> However, this does *not* imply that > > >> there are two numbers that are not in a single line. > > > > > Why then should two lines be necessary? > > > One being the substitute in case the other falls ill? > > > > >> Nor does it imply that there is a necessary line in K. > > > > > If there is not one necessary line, then there are two or more > > > required. > > > Proof: If you remove all lines from the list, then there remains no > > > line and no number. > > > > >> Note that a sufficient set does not imply a necessary line > > >> even in potential infinity. There is no line that is needed > > >> to make L have an unfindable last line. > > > > > So you believe that there can remain all numbers in the list after > > > removing all lines? That is a remarkable claim. I would not accept it > > > in mathematics. > > > > > Note in actual infinity it makes sense to talk about all lines and to > > > remove all lines. > > > > I think this is just a difference of interpretation > > concerning "necessary". > > That is easily clarified: How many lines can be removed without > removing one of the natural numbers of the list.
If that "list" is the union of a finite set of lines then all but the last/longest line for that finite set of lines.
And all but infinitely many from any infinite set of lines. Note that one can always remove infinitely many more lines from an infinite set of lines as long as one leaves infinitely many without affecting the union of the lines left. > > > > To borrow from linear algebra, you are describing > > something that might be more along the lines of > > a "spanning set". > > No. In my case every line is the super set of all preceding lines. > Therefore we have a linear ordering.
WM is not in a good position to pontifice about linearity while he still has his "linear mapping" of binary sequences problem to solve. > > But of course we have to apply some fundamental rules. For instance we > must believe that removing all lines will remove all numbers from the > list. Of course in matheology also the contrary belief is admissible.
The only place I have seen anything that wrong recently is in Wolkenmuekenheim.
> For instance, removing all lines removes only all nameable numbers. If > someone believes in non-nameable numbers, we cannot prove that the > list is empty.
WM must believe in non-nameable naturals, as his sets of naturals always end with some natural which one cannot either name or exceed. So that one, and all its followers are unnameable in Wolkenmuekenheim.
WM has frequently claimed that a mapping from the set of all infinite binary sequences to the set of paths of a CIBT is a linear mapping. In order to show that such a mapping is a linear mapping, WM must first show that the set of all binary sequences is a vector space and that the set of paths of a CIBT is also a vector space, which he has not done and apparently cannot do, and then show that his mapping satisfies the linearity requirement that f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of a field of scalars and x and y are f(x) and f(y) are vectors in suitable linear spaces.
By the way, WM, what are a, b, ax, by and ax+by when x and y are binary sequences?
If a = 1/3 and x is binary sequence, what is ax ? and if f(x) is a path in a CIBT, what is af(x)?
Until these and a few other issues are settled, WM will still have failed to justify his claim of a LINEAR mapping from the set (but not yet proved to be vector space) of binary sequences to the set (but not yet proved to be vector space) of paths ln a CIBT.
Just another of WM's many wild claims of what goes on in his WMytheology that he cannot back up. --