Date: Jan 26, 2013 10:06 AM Author: Jesse F. Hughes Subject: Re: ZFC and God WM <mueckenh@rz.fh-augsburg.de> writes:

> 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?

You've snipped the question three times, in the thread directly

preceding this post.

> 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.)

I don't know why you want to avoid using the usual convention that

0.1 = 0.1000...., but okay. It makes no difference.

Let's state the definition explicitly then:

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.

Right?

Now, let {t_i} be a list of all the finite decimal representations of

reals, that is, each t_i is a finite decimal representation, and every

finite decimal representation is in the list. For each t_i, let k_i

be the "length" of t_i.

And we define a sequence d_j so that

d_j = 7 if j > k or t_j(j) != 7

d_j = 6 if j <= k and t_j(j) = 7.

As before, we can notice the following facts:

d_j is defined for every j in N.

d_j = 7 or d_j = 6 for every j in N.

Clearly, d_j is *NOT* a finite sequence. Moreover, since the sequence

d_j does not end in trailing 0s or 9s, the real number d defined by

d = sum_i=1^oo d_i & 10^-i

has no finite decimal representation.

Now, please tell me what is unclear about these obvious facts?

--

Jesse F. Hughes

"Mistakes are big part of the discovery process.

I make lots of them. Kind of pride myself on it."

-- James S. Harris