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.