In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 20 Mrz., 14:00, William Hughes <wpihug...@gmail.com> wrote: > > On Mar 20, 1:17 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > <snip> > > > > > > So your proof that any two lines can be replaced > > > > by one line without changing the contents is irrelevant. > > > > > Since contents can only exist in lines, and since every line is > > > superset to all its predecessors, the proof is correct. > > > > The proof is irrelevant (it is, however, correct) > > Nice to hear. Not that I had any doubt, but it is nice to hear that > you have no doubt too. > > > since showing that lines are not needed for their > > contents does not show that the lines are not needed. > > The lines were invented by myself solely for this purpose. Then the invention was futile for your purpose was not achieved, as at least infinitely many of those lines are necessary and have been proved to be. > > The first question is: > Is the first line necessary to have the number 1 in the list. > > Formulated somewhat more "mathematically": > Is the union of all lines different from the union of all lines except > the first one.
Theorem: Every finite union of lines omits some naturals and every infinite union of lines includes all naturals. > > Then go to the second question. > > > To show that actual infinity is unreasonable you > > have to show the lines are not needed. > > I show that every line, that is not the last line, is not needed.
Wrong! That implies a condition contrary to fact. But one can show that every line that is not A last line is not needed.
> Since there is not a last line in actual infinity, this shows that all > lines are not needed.
Since the union of all infinitely many lines/FISONs together is clearly sufficient to contain all naturals, there may be some minimal condition on a union of lines which is a necessary condition, and that minimal condition is that the set of lines/FISONs to be unioned is infinite.
And WM has still not validated his claim for existence of a linear mapping from the of all infinite binary sequences onlt the set of all paths of Complete Infinite Binary Tree. A bjection is trivial, a truly linear mapping is well beyond WM's skills. --