Date: Jan 26, 2013 4:36 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

On 26 Jan., 02:50, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> >> I'm not going to bother working through your addled analogy.
>
> > You need not. Just ask yourself whether or not it is possible to
> > define in ZFC the set of all terminating decimal representations of
> > the real numbers of the unit interval. If you think that it is not
> > possible, then you should try to learn it. If you know it already,
> > then we can formally restrict ourselves to working in this set until
> > we discover a digit that is not defined in an element of this set.

>
> > Your further questions then turn out meaningless.
>
> I asked how you define terminating decimal representation.  How is
> that meaningless?


Sorry, where did you ask?
Nevertheless, the answer is: A terminating decimal representation
(0.d_1,d_2,..., _n) has a finite set of indices {1, 2, ..., n} with n
a natural number.
>
> Here's the definition I suggested again.  Please tell me if you agree
> with it, and if not, what definition you have in mind.
>
> Let x be a real number in [0,1].  We say that x has a terminating
> decimal representation iff there is an f:N -> {0,...,9} such
> that
>
>   x = sum_i f(i) * 10^-i,
>
> and
>
>   (En)(Am > n)(f(m) = 0) or (En)(Am > n)(f(m) = 9)


The latter is not quite correct, because a terminating decimal
representation has nothing behind its last digit d_n, neither zeros
nor any other digits. (But of course, we can expand every terminating
decimal by a finite set of further decimals d_j = 0 for every j with n
<j <m, m in N.)
>
> If x has no terminating decimal representation, then we say that x is
> non-terminating.
>

Yes.

> We cannot continue unless I know what you mean by terminating decimal
> representation.


Should be clear now.

Regards, WM