Date: Jan 24, 2013 11:02 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

On 24 Jan., 14:16, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 24 Jan., 13:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> >> WM <mueck...@rz.fh-augsburg.de> writes:
>
> >> Well, what you present below is *not* a proof of (*).
>
> > That is wrong. You have no reason to believe that your definition of
> > proof is correct or the only one.

>
> This argument doesn't involve U_n {1,...,n} *at all*!
>
> But let's let it pass.


If you can't understand that we have (d_1, ..., d_n) with just all the
indices given by the FIS above, then we should stop for a while.


> > 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.
>
>


> > Neither the set of t_i does have a largest element. Nevertheless there
> > is no t_i of actually infinite length.

>
> Correct, to both claims. So what? The digit d(j) is defined for
> every j, and is neither 0 nor 9, so d is a real number which has no
> terminating decimal representation.
>
> Do you agree with that (obvious) claim or not?


No. Pleeze try to understand: Presently we work in the system of
terminating decimals, by definition. There is no non-terminating
decimal. If we were so unlucky to met any non-terminating decimal we
would not know what to do. It would not even be defined. If the
diagonal gets non-terminating, we have to stop and cut it before it
becomes non-terminating.

>
> Sorry, I could've sworn you were talking about what ZF proves. Can
> you state this in the language of ZF and prove it? Much thanks!


Can't you imagine to define in ZF the set of all terminating decimals?
Is it too hard? Just take the above set that you couldn't recover in
our arguing, namely the set of all FIS (1, 2, ..., n) with n in n as
indices of the digits.

> > In particular, what would be changed in the length of d if we admitted
> > also non-terminating t_i (of infinite length)?

>
> Nothing would change. So what?


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?

Regards, WM