In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 30 Jan., 10:52, William Hughes <wpihug...@gmail.com> wrote: > > On Jan 30, 10:46 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > > > > > > > On 30 Jan., 10:31, William Hughes <wpihug...@gmail.com> wrote: > > > > > > For a potentially infinite list L, the > > > > antidiagonal of L is not a line of L. > > > > > Of course. Every subset L_1 to L_n can be proved to not contain the > > > anti-diagonal > > > > > > Does this imply > > > > > > There is no potentially infinite list > > > > of 0/1 sequences, L, with the property that > > > > any 0/1 sequence, s, is one of the lines > > > > of L. > > > > > Do you mean potentially infinite sequences? > > > > yes- > > A potentially infinite sequence has *not* more elements than every > natural number.
In standard mathematics, sets are either actually finite or actually not finite (infinite). Tertium Non Datur.
A set is by definition finite provided there are no injections from it to any of its proper subsets and not finite (infinite) if at least one such injection exists. Thus the mapping n -> n+1 proves |N is not finite according to this defnition.
While it may not be known, or even knowable, which holds for a particular set, in standard math it must be one of those two possibilities. Tertium non datur. --