```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 thecomplete 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 areworking in the set of terminating decimals and identify numbers bytheir 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 isnot in the set of all terminating decimals.Regards, WM
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