In article <d_OdnaqCGY6gO9HMnZ2dnUVZ_uidnZ2d@giganews.com>, fom <fomJUNK@nyms.net> wrote:
> On 3/22/2013 1:21 PM, WM wrote: > > On 22 Mrz., 16:31, William Hughes <wpihug...@gmail.com> wrote: > >> On Mar 22, 10:05 am, WM <mueck...@rz.fh-augsburg.de> wrote: > >> > >>>> If you want to > >>>> remove all of the lines you have to remove the set of all > >>>> lines that are indexed by a natural number. > >> > >>> But I don't want to remove a set. > >> > >> We have the set of lines. You do not want to leave > >> any of the lines. > > > > I do not want this or that. > > I simply prove that for every line l_n the following property is true: > > Line l_n and all its predecessors do not in any way influence (neither > > decrease nor increase) the union of all lines, namely |N. > > > > This is certainly a proof that does not force us to "remove a set". > > But we can look at the set of lines that have this property. The > > result is the complete set of all lines. > > So now you have two complete infinities. > > The infinity of objects comprising |N. > > The infinity of finite lines whose sequentially > ordered elements are from |N. > > Which one is *the* infinity? > > Or, if there are many, how do they stand > in relation to one another?
One way out would be to use the Von Neumann zero-origin model for the naturals, with the first natural, 0, being the empty set and each subsequent natural being the set of all previous ones.
With this model, each natural. except possibly 0. could also be its own line, and we would no longer have to distinguish between naturals and lines.
WM only objects to vN naturals because they diminishe his abiity to obscure his ubiquitous illogic. --