In article <aa51345c-20b8-4e58-92b5-39575aa1eca9@q4g2000yqh.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 13 Okt., 23:13, William Hughes <wpihug...@gmail.com> wrote: > > On Oct 13, 12:39 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > > > > > > > On 13 Okt., 15:29, William Hughes <wpihug...@gmail.com> wrote: > > > > > > So we are agreed that you can have a finished infinity > > > > of symbols that contains a symbol for every natural number but does > > > > not contain the symbol oo. (note that this finished infinity > > > > does not have a largest element). > > > > > Of course, if there are no other arguments contradicting it. > > > > > > The first column contains every natural number > > > > but does not contain oo, and the diagonal contains every natural > > > > number but does not contain oo, They are identical. > > > > > And here is the argument contradicting it: > > > The third side must also contain every natural number, but does not. > > > > There is no third side. > > > > A third side would have to be the last line. > > Then you could argue: There is no first side. A complete first side > would have to have a last element. > > > > The fact that a limit, L, of things from the triangle > > exists does not mean that L is part of the triangle. > > Correct! Therefore also the other "complete" edges are not part of the > triangle. And 1/9 is not is not part of the triangle > > 0.1 > 0.11 > 0.111 > ... > > neither in a line nor in the right edge.
When there is a last line to your sequence, on can regard the "1"'s as forming a right triangle, but when there is no last line, there is also no third side, so no triangle. > > Cantor's "proof" rests upon the naive assumption that 1/9 must be the > right edge because the number of 1's there form a fixed number.
Actually WM has it backwards, as usual. If the diagonal right edge does not terminate then one does have an endless sequence of 1's and the value of the expression does equal 1/9.
> Obviously that is the same number as the number of lines and the > number of columns.
But that number, unless there is a last line and a last column, is not a finite number. > > I proved
Not to the satisfaction of anyone other than yourself! --
