Date: Jan 27, 2013 7:10 AM
Author: Jesse F. Hughes
Subject: Re: ZFC and God

WM <mueckenh@rz.fh-augsburg.de> writes:

> On 26 Jan., 23:19, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>

>> > It is unclear why you apparently are unable to understand, that we are
>> > working in the set of terminating decimals. Therefore the diagonal
>> > cannot be actually infinite, although there is no last digit.

>>
>> Let me ask you a very simple question.
>>
>>   Is 0.777.... a terminating decimal representation or a
>>   non-terminating decimal representation?

>
> That depends on the domain where you work in. We have started to work
> in the domain of terminating decimals. Since the diagonal consists
> only of (changed) digits of these decimals, it is obviously a
> terminating decimal.
> Now, to answer your question: You did not say where you take 0.777...
> from. And obviously that cannot be determined from the digits, as I
> jusr explained.


When I write 0.777..., I mean the number

sum_i=1^oo 7 * 10^-i

That is, for each i in N, the i'th digit of 0.777... is defined and is
7.

Do you agree that there is only one number satisfying that
description? Or are there two numbers that satisfy that description
and one of the numbers is terminating and the other non-terminating?

Let's suppose there *are* two different numbers, corresponding to the
terminating 0.777... and the non-terminating 0.777... . Then

term. 0.777... = sum_i=1^oo 7*10^-i

and also

non-term. 0.777... = sum_i=1^oo 7*10^-i,

but then, of course, term. 0.777... = non-term. 0.777... ! Oops!

Moreover, neither term. 0.777... nor non-term 0.777... satisfy the
definition of terminating decimal that you previously agreed to,
namely

Let x be a real number in [0,1].  We say that x has a terminating
decimal representation iff there is a natural number k and a
function f:{1,...,k} -> {0,...,9} such that


  x = sum_i=1^k f(i) * 10^-i.

The "terminating" 0.777... has no finite length.

--
"I liked the world a lot better over ten years ago. I believed in a
lot more things. Hell, most people believed in a lot more things.
Back then the United States was still, well, known as most people used
to know the United States." -- James S. Harris in a nostalgic mood