> 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.
You're throwing about terms (potential, restricted to the domain..., etc.) that have no obvious meaning in the theory ZF, so I see no reason to reply to this. We're trying to prove a certain claim in ZF, so unless you can indicate how to interpret these terms, I have nothing to say. > >> >> 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?
Is t_i(i) also defined for every i in N?
Assuming you will say yes, then I must ask:
Is d(i) therefore defined for every i in N?
>> 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?
If they have any meaning in the language of ZF, then simply tell me what the meaning is.
-- "I suggest to those who listen that they enjoy the world, whatever their piece of it may be, as much as they can over the next few days, as soon enough, it will pass away, thanks to people who call themselves 'mathematicians'." -- JSH envisions geek Ragnarok