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