Date: Jan 27, 2013 4:54 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

On 27 Jan., 19:56, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 27 Jan., 19:21, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> >> WM <mueck...@rz.fh-augsburg.de> writes:
> >> > On 27 Jan., 18:44, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> >> >> Anyway, you haven't proved that there is a function
>
> >> >>   f:{1,...,k} -> {0,...,9}
>
> >> >> as required by *your* definition of terminating decimal, so you have
> >> >> not shown that 0.777... is a terminating decimal.

>
> >> > You are wrong. Can't you understand? All natural numbers are finite.
> >> > Why the heck should I define a single k?

>
> >> Because, of course, you accepted the following definition:
>
> >>   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.
>
> > I did not fix k but only assumed that it is a natural number.
>
> Pardon me?
>
> Look, you said that a number has terminating decimal representation
> iff there is some k and f such that blah blah blah.  Thus, to show
> that a number *does* have a terminating decimal representation is
> equivalent to showing that there is, in fact, a k and f satisfying the
> above.


Of course. But why should we agree on a special k? Every natural
number will do. So we only have to know that k is one of those natural
numbers that belong to FISONs. As long as we work in FISONs we cannot
have a non-terminating decimal.
>
> What could be more obvious than this?


Remember the Binary Tree to answer your doubts. Do you believe that
the set of all FISONs is definable in ZF? Is every FISON finite? Is
there a fixed k limiting all FISONs? No.
>
> > You have no cause to claim the contrary, since there is no index
> > (natural number) infinitely many counts away from the decimal point.

>
> Are you asking me to prove that 0.777... does not have a terminating
> decimal representation?  I would be happy to do so, if needed.


I am doubting that when working in FISONs you can find an index of a
diagonal number that requires to leave the set of FISONs (with respect
to the indexes of the digits).

> it follows *FROM THE AGREED DEFINITION* that 0.777... has no
> terminating decimal representation.


Show that in your 0.777..., and in particular in the anti-diagonal of
a list of terminating decimals, there is an index k that does not
belong to a FISON {1, 2, ..., n}. As long as you refuse there is no
reason to believe you, because every index in a FISON belongs to a
FISON (finite initial set or sequence of natural numbers). Why do you
try to deny that?

Regards, WM