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 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.