```Date: Jan 25, 2013 2:52 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

On 25 Jan., 01:39, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:> WM <mueck...@rz.fh-augsburg.de> writes:> > On 24 Jan., 14:16, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:> >> WM <mueck...@rz.fh-augsburg.de> writes:> >> > You will have have recognized that here the diagonal argument is> >> > applied. It is obvious that up to every line = column the list is a> >> > square.>> >> It is clear that, for all j, d(j) != t_j(j) and hence d != t_j.  If> >> that's what you mean by the diagonal argument, great!>> >> Once again, however, you say something that has no clear meaning to> >> me.  Can you clarify "It is obvious that up to every line = column the> >> list is a square?"  I've no clue what it means.>> > Then ponder a while about the following sequence>> > d>> > d1> > 2d>> > d11> > 2d2> > 33d>> > and so on. In every square there are as many d's as lines. The same> > could be shown for the columns.>> Yes, in this sequence of three squares, what you say is true.Is there a first square where my observation would fail?>> But none of this is relevant, because we've explicitly defined the> anti-diagonal d and it is a triviality to see that it is an infinite> sequence of non-zero and non-nine digits.  And this fact really has> nothing at all to do with limits of sequences of squares.  It is all> perfectly explicit.Here you again intermingle potential and actual. We are restricted tothe domain of terminating decimals. If you cannot understand that,perhaps a formal argument may help: Assume that we are restricted tothe well-defined set of terminating decimals. If you see any evidencethat we should leave that domain, say "stop!". But only if you aresure.>> Do you agree that (by presumption) t_i is defined for every i in N?Of course! Why not? Isn't every i in N finite?>> I don't want to imagine what you are thinking, because I will risk> getting it wrong.  I'd prefer that you explicitly give an argument in> ZF so that we can determine whether it is valid or not.In ZF every n in N is finite.>> > Look, presently we work in the system of terminating decimals - by> > definition. If nothing changes when we switch to the system of non-> > terminating decimals, do we switch then at all? How could we recognize> > that we have switched?>> I don't have any idea what these questions meanI know. But it would be nice if you read it again and again. Or try anexperiment: Write a long sequence of digits d_1, d_2, d_3, ... and donot stop. Are you in danger to leave the domain of finite sequences?Regards, WM
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