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Re: ZFC and God
Posted:
Jan 26, 2013 5:19 PM
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WM <mueckenh@rz.fh-augsburg.de> writes:
> On 26 Jan., 16:06, "Jesse F. Hughes" <je...@phiwumbda.org> 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.
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