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. > > 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.
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. 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. (This given only as one example of how matheologians would argue to remain with their belief - which is the regula prima anyhow).