Date: Jan 27, 2013 8:46 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

On 27 Jan., 13:10, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> WM <mueck...@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.


And do you have problems to find this confirmed as possible in the
complete set of terminating decimals? Any digit or index missing?
>
> 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?

I agree that this is a finite definition. But I said that we are
working in the set of terminating decimals and identify numbers by
their digits, indices or nodes. Is that hard to understand?
>
> 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.

Please let me know when you will have succeded in finding a 7 that is
not in the set of all terminating decimals.

Regards, WM