On 24 Jan., 13:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > WM <mueck...@rz.fh-augsburg.de> writes:
> Well, what you present below is *not* a proof of (*).
That is wrong. You have no reason to believe that your definition of proof is correct or the only one.
> > Clearly, for all j, d(j) != t_j(j) and hence d != t_j for any j in > N. > > Is this what you mean up 'til now?
> > > 4) Certainly you agree that, since all t_i = (t_i1, t_i2, ..., t_in) > > have only a finite, though not limnited, number n of digits, the > > diagonalization for every t_i yields a finite d_i =/= t_ii. > > (The i on the left hand side cannot be larger than the i on the right > > hand side. In other words, "the list" is a square. Up to every i it > > has same number of lines and columns. ) > > No idea what you mean by the parenthetical remark.
You will have have recognized that here the diagonal argument is applied. It is obvious that up to every line = column the list is a square. > > I do agree that d_i is defined for every i in N. In particular, (d_i) > is an infinite sequence of digits. Is this what you're claiming, too? > You've lost me. I don't know what you mean when you say, "everything > here happens among FISs." And I'm also puzzled by the meaning of the > next sentence.
Every t_i is finite. Hence, in a square, if the width is finite, also the length must be finite. > > Here are some obvious things. > > d(j) is defined for every j in N. > d(j) != 0 and d(j) != 9 for any j in N. > > Hence the number d does not have a terminating decimal > representation.
Neither the set of t_i does have a largest element. Nevertheless there is no t_i of actually infinite length. > > This looks like I do *not* agree with your claim that "d cannot be > longer than every t_i".
A sequence of squares will never result in a square such that all sides are finite but the diagonal d is infinite. The overlap of d and t_i cannot be larger than t_i.
In particular, what would be changed in the length of d if we admitted also non-terminating t_i (of infinite length)?