Date: Jan 24, 2013 7:56 AM
Author: Jesse F. Hughes
Subject: Re: ZFC and God

"Jesse F. Hughes" <jesse@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.

--
Jesse F. Hughes
"Well, if I can get [my proof of FLT accepted], then I hopefully get a
book deal down the road, and maybe I get to go on 'Oprah'."
James Harris, on the rewards of mathematical endeavours.