```Date: Jan 27, 2013 3:59 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

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 workin the domain of terminating decimals. Since the diagonal consistsonly of (changed) digits of these decimals, it is obviously aterminating 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 Ijusr explained.I could answer:  You can look whether there is a digit of 0.777...that is not in a (in fact in infinitely many) finite initialsegment(s). Then you have a proof that 0.777... does not belong to theset of terminating decimals. But it is clear that you cannot find sucha digit. Therefore you can only decide your question by defining where0.777... has been talen from. The reason for this uncertainty is thefact, that the Binary Tree constructed by all finite paths cannot bedistinguished by digits (i.e. without further definition) from thatBinary Tree that contains all infinite paths too.>>   Does the real number corresponding to 0.777.... have a terminating>   decimal representation?>> Much thanks for answer what is surely a trivial question.You are in error. The question unfortunately is far from beingtrivial. It has only been overlooked that the sets F of all finitedecimals and R of all infinite decimals cannot be distinguished otherthan by a finite definition.Would be nice if you really tried to understand that, although itrequires a complete change of your habbits of thinking. But if youconsider the different Binary Trees, you should come to the correctresult.Here a trivial example: The set of all finite initial segments of theinfinite path of 1/3 = 0.010101... contains all nodes that belong tothat path. So you cannot distinguish *by nodes* whether you areworking in the set F or the set R. You need additional information.But that informtation does not matter in Cantor's argument.Regards, WM
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