Date: Jan 25, 2013 2:52 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

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?

Regards, WM