Date: Jan 26, 2013 5:19 PM
Author: Jesse F. Hughes
Subject: Re: ZFC and God

WM <> writes:

> On 26 Jan., 16:06, "Jesse F. Hughes" <> wrote:
>> 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?

> 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?

> 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?

Does the real number corresponding to 0.777.... have a terminating
decimal representation?

Much thanks for answer what is surely a trivial question.

"[I]f I could go back, [...] I would tell myself not to step into a position
where the fate of the entire world could rest in my hands. I would [avoid
this] path to a nightmarish and surreal world, a topsy-turvy world, where
everything changes." -- James S. Harris cannot escape his destiny.