```Date: Jan 27, 2013 7:10 AM
Author: Jesse F. Hughes
Subject: Re: ZFC and God

WM <mueckenh@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^-iThat is, for each i in N, the i'th digit of 0.777... is defined and is7.Do you agree that there is only one number satisfying thatdescription?  Or are there two numbers that satisfy that descriptionand one of the numbers is terminating and the other non-terminating?Let's suppose there *are* two different numbers, corresponding to theterminating 0.777... and the non-terminating 0.777... .  Then   term. 0.777... = sum_i=1^oo 7*10^-iand 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 thedefinition 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.-- "I liked the world a lot better over ten years ago. I believed in alot more things. Hell, most people believed in a lot more things.Back then the United States was still, well, known as most people usedto know the United States." -- James S. Harris in a nostalgic mood
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