Date: Jan 24, 2013 8:29 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

On 24 Jan., 13:56, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> "Jesse F. Hughes" <je...@phiwumbda.org> writes:
>

> > Yes, but let's be perfectly clear here.  What we have now is this:
>
> >   a function t: N -> R such that, for all i in N, t(i) is a real
> >   number with terminating decimal representation.  We'll write this as
> >   t_i and we'll write t_i(j) for the j'th digit of t_i (in its
> >   terminating representation, which is, of course, unique).

>
> >   Let d in R be defined by
>
> >     d(j) = 7 if t_j(j) != 7
> >     d(j) = 6 if t_j(j) = 6.

>
>                            ^ Should be 7, of course.
>
>
>

> >   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?
>
> So, for instance, take the obvious enumeration of finite decimal
> representations (in [0,1), of course)
>
> 0.0
> 0.1
> 0.2
> 0.3
> 0.4
> 0.5
> 0.6
> 0.7
> 0.8
> 0.9
> 0.11
> 0.12
> ...
>
> Clearly, t_i(i) = 0 for every i, and hence d_i = 7 for every i.  Thus,
> d = 0.777..., which is a number without any finite decimal
> representation.


As long as the line has a finite enumeration, also the column has. In
t_j(j) the first j is identical with the second j. And both belong to
a FIS of lines and columns, respectively. This holds for every j.

Regards, WM