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