Date: Jan 26, 2013 10:06 AM
Author: Jesse F. Hughes
Subject: Re: ZFC and God

WM <> writes:

> On 26 Jan., 02:50, "Jesse F. Hughes" <> wrote:
>> WM <> 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.


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