Date: Jan 23, 2013 11:02 AM Author: mueckenh@rz.fh-augsburg.de Subject: Re: ZFC and God On 23 Jan., 14:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> WM <mueck...@rz.fh-augsburg.de> writes:

> > I know. But if you have read the discussion, you have seen that two

> > matheologians claim just this. Why do they? Because they cannot answer

> > the question: What paths are (as subsets of the set of nodes) in a

> > Binary Tree that is the union of all its levels? Are there only the

> > finite paths? Or are there also the infinite paths?

> > Try to answer it, and you will see that you need the omegath level or

> > must confess that it is impossible to distinguish both cases. Hence,

> > Cantor's argument applies simultaneously to both or to none.

>

> I'm not interested in the web-published claims of two individuals on a

> different topic than we're discussing.

You are in error. Pause for a while and think it over.

>

> Once again, let me remind you what you claimed. You claimed ZF was

> inconsistent, and in particular that ZF proves that the union

>

> U_n {1,...,n}

>

> is both finite and infinite.

>

> Now, we've had two competing definitions of infinite in this

> particular discussion.

>

> (1) A set S is infinite if there is no natural n such that |S| = n.

>

> (2) A set S is infinite if it contains a number greater than every

> natural n.

>

> The first definition is what mathematicians almost always mean, and

> they *never* mean the second, but this is mere semantics. Let's talk

> results.

You are right, mathematicians prefer (1). But matheologians use (2).

An infinite set contains a number of elements, at least aleph_0, which

is greater than every finite number.

>

> We both agree that, using definition (1), the above union is infinite

> and (I think) we agree that we cannot show it is finite (=not

> infinite). If I'm mistaken on this point, then please show me.

>

> On the other hand we both agree that, per definition (2), the union is

> "finite", but I have seen no contradiction result, since you have not

> shown that the union is "infinite" in this sense. Nor can you find a

> single publication in which a mathematician has claimed the union

> above (i.e., the set N of natural numbers) contains an element larger

> than every natural.

You confuse the things. ZF claimes that the *number of elements* is

larger than every finite number. Just this causes the contradiction. A

union of finite initial segments cannot have a number of elements that

is larger than every finite number.

Regards, WM