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".
To borrow from linear algebra, you are describing something that might be more along the lines of a "spanning set". Any particular lines are not necessary, but whenever one speaks of the possibility of given lines containing all the numbers, the count of those lines would necessarily have a non-zero value greater than one because of partiality.