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Re: ZFC and God
Posted:
Jan 27, 2013 1:33 PM
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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.
>
> > 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.
If you insist in a non-terminating one, please show it!
Regards, WM
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