Date: Jan 24, 2013 7:09 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

On 24 Jan., 12:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 23 Jan., 19:04, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> >> > Understand the Binary Tree. After you will have understood it, you
> >> > will understand, why it is important.

>
> >> No, let's first settle the point at hand.
>
> > I do it by means of tools that I choose without any censorship from
> > your side.

>
> I'm not being unreasonable here.  You say ZF is inconsistent.  I want
> to see whether you can indeed show that.
>
> So, I'd like to know what inconsistency you can show in ZF.  You have
> already said (don't let me put words in your mouth!  Correct me if I'm
> wrong) that you can show ZF proves
>
>   U_n {1,...,n} is not infinite.             (*)


It is not actually infinite. The cardinality is not larger than every
n.
>
> Since we know it also proves the negation of (*), this would settle
> your claim.
>
> Now, I'd just like to see the proof of (*).  Nothing else.  Just show
> me that proof and we'll discuss it.


Ok.

1) Certainly you agree that in ZF we have the set T of all terminating
decimal fractions t_i of the reals in the unit interval, i.e., finite
sequences of digits, indexed by the FISs {1,...,n}.

2) Certainly you agree that the set T is countable.

3) Certainly you agree that the set can be diagonalized.

4) Certainly you agree that, since all t_i = (t_i1, t_i2, ..., t_in)
have only a finite, though not limnited, number n of digits, the
diagonalization for every t_i yields a finite d_i =/= t_ii.
(The i on the left hand side cannot be larger than the i on the right
hand side. In other words, "the list" is a square. Up to every i it
has same number of lines and columns. )

So everything here happens among FISs. And d cannot be longer than
every t_i. Nevertheless d_i differs from every t_ii. So we see that ZF
proves the uncountability of a countable set.

Regards, WM