On 5 Apr., 21:03, William Hughes <wpihug...@gmail.com> wrote: > On Apr 5, 6:04 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 5 Apr., 12:08, William Hughes <wpihug...@gmail.com> wrote: > > <snip> > > > > There is an infinite set of lines D > > > such that any finite subset of D can be removed. > > > What has to remain? > > This depends on the finite subset removed. > If the finite set removed is E then > D\E has to remain. Note that whatever > subset E is chosen the number of lines > in D\E is infinite
How do you call a set E the number of elements exceeds any given natural number? (You do not claim that we can only remove a set E with less than a given natural number, do you?)
> (but of course we > do not know which lines are in D\E).
How do we call a set when we cannot biject it with a FIS on |N?
> However, D cannot be removed without > changing the union of the remaining lines.
That is correct, if D is not more than the union of all finite lines.
But if so, then every Cantor list that contains all rational numbers has the following property:
For every n in |N: There are infinitely many lines that have the same finite initials sequence d_1, d_2, d_3, ..., d_n of digits as the anti- diagonal.
Only if the diagonal is more than every FIS, i.e., the list is more than every finite lines, then this proof could be objected. For all n in |N in is valid.
So you are caught in a circulus vitiosus: Either actual infity D is more than all its FISs, then all can be removed without changing which is obviously nonsense, or D is not more, then there is a proof against Cantor's theorem which is as valid as Cantor's proof.