Date: Jan 27, 2013 4:53 PM
Author: Virgil
Subject: Re: ZFC and God

In article 
<653bcb61-ebf2-4fd6-a852-3de91cc792bf@v5g2000yqg.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> 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.

> >
> > Thus, if you claim that 0.777... has a terminating representation,
> > then you must show that there is a natural number k and a function f
> > as above such that
> >
> >   0.777... = sum_i=1^k f(i) * 10^-i.
> >
> > Else, you have no cause to claim that 0.777... has a terminating
> > decimal representation.
> >

> You have no cause to claim the contrary, since there is no index
> (natural number) infinitely many counts away from the decimal point.



Either there is a specific K in |N such that
0.777... = sum_i=1^k f(i) * 10^-i
or there is no such k.

If there is one then 0.777... = 0.777...7770 for some 0.777...7770.
> >
> > > This is the definition that I agreed to.
>
> >
> > Frankly, I'm a bit stunned that you're arguing that 7/9 has a
> > terminating decimal representation, but as long as you're claiming so,
> > then you need to stick to the definition we've agreed on.

>
> I am not claiming that 7/9 ot 1/3 or sqrt(2) have decimal
> representations at all.
> Just the contrary. But I am claiming that all decimal representations
> that exist in the domain of terminating decimals are terminating, in
> particular the diagonal of a Cantor-list, as long as we work in the
> domain of terminating decimal representations.


Bu you cannot provide any good enough reason why, when not in
Wolkenmuekenheim, one should work with only terminating decimal
representations when standard analysis allows infinite series.
>
> If you insist in a non-terminating one, please show it!


7/9 = sum_(i in |N) f(i) * 10^-i

Since standard analysis allows infinite series, there is no eeason why
we need avoid them here.
>
> Regards, WM

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