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

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 positionwhere the fate of the entire world could rest in my hands.  I would [avoidthis] path to a nightmarish and surreal world, a topsy-turvy world, whereeverything changes." -- James S. Harris cannot escape his destiny.
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