On 25 Jan., 01:39, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > WM <mueck...@rz.fh-augsburg.de> writes: > > On 24 Jan., 14:16, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > >> WM <mueck...@rz.fh-augsburg.de> writes: > >> > 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. > > >> It is clear that, for all j, d(j) != t_j(j) and hence d != t_j. If > >> that's what you mean by the diagonal argument, great! > > >> Once again, however, you say something that has no clear meaning to > >> me. Can you clarify "It is obvious that up to every line = column the > >> list is a square?" I've no clue what it means. > > > Then ponder a while about the following sequence > > > d > > > d1 > > 2d > > > d11 > > 2d2 > > 33d > > > and so on. In every square there are as many d's as lines. The same > > could be shown for the columns. > > Yes, in this sequence of three squares, what you say is true.
Is there a first square where my observation would fail? > > But none of this is relevant, because we've explicitly defined the > anti-diagonal d and it is a triviality to see that it is an infinite > sequence of non-zero and non-nine digits. And this fact really has > nothing at all to do with limits of sequences of squares. It is all > perfectly explicit.
Here you again intermingle potential and actual. We are restricted to the domain of terminating decimals. If you cannot understand that, perhaps a formal argument may help: Assume that we are restricted to the well-defined set of terminating decimals. If you see any evidence that we should leave that domain, say "stop!". But only if you are sure.
> > Do you agree that (by presumption) t_i is defined for every i in N?
Of course! Why not? Isn't every i in N finite? > > I don't want to imagine what you are thinking, because I will risk > getting it wrong. I'd prefer that you explicitly give an argument in > ZF so that we can determine whether it is valid or not.
In ZF every n in N is finite. > > > Look, presently we work in the system of terminating decimals - by > > definition. If nothing changes when we switch to the system of non- > > terminating decimals, do we switch then at all? How could we recognize > > that we have switched? > > I don't have any idea what these questions mean
I know. But it would be nice if you read it again and again. Or try an experiment: Write a long sequence of digits d_1, d_2, d_3, ... and do not stop. Are you in danger to leave the domain of finite sequences?