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