Date: Jan 27, 2013 8:46 AM Author: mueckenh@rz.fh-augsburg.de Subject: Re: ZFC and God On 27 Jan., 13:10, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> WM <mueck...@rz.fh-augsburg.de> writes:

> > On 26 Jan., 23:19, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

>

> >> > 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?

>

> > That depends on the domain where you work in. We have started to work

> > in the domain of terminating decimals. Since the diagonal consists

> > only of (changed) digits of these decimals, it is obviously a

> > terminating decimal.

> > Now, to answer your question: You did not say where you take 0.777...

> > from. And obviously that cannot be determined from the digits, as I

> > jusr explained.

>

> When I write 0.777..., I mean the number

>

> sum_i=1^oo 7 * 10^-i

>

> That is, for each i in N, the i'th digit of 0.777... is defined and is

> 7.

And do you have problems to find this confirmed as possible in the

complete set of terminating decimals? Any digit or index missing?

>

> Do you agree that there is only one number satisfying that

> description? Or are there two numbers that satisfy that description

> and one of the numbers is terminating and the other non-terminating?

I agree that this is a finite definition. But I said that we are

working in the set of terminating decimals and identify numbers by

their digits, indices or nodes. Is that hard to understand?

>

> Let's suppose there *are* two different numbers, corresponding to the

> terminating 0.777... and the non-terminating 0.777... . Then

>

> term. 0.777... = sum_i=1^oo 7*10^-i

>

> and also

>

> non-term. 0.777... = sum_i=1^oo 7*10^-i,

>

> but then, of course, term. 0.777... = non-term. 0.777... ! Oops!

>

> Moreover, neither term. 0.777... nor non-term 0.777... satisfy the

> definition of terminating decimal that you previously agreed to,

> namely

>

> 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.

>

> The "terminating" 0.777... has no finite length.

Please let me know when you will have succeded in finding a 7 that is

not in the set of all terminating decimals.

Regards, WM