Date: Jan 23, 2013 2:26 PM
Author: Virgil
Subject: Re: ZFC and God

In article 
<9b8527a8-fabc-49f0-91bc-2cb110b9b570@f4g2000yqh.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> 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).


I have never seen anyone, other than WM himself, use 2, and have
certainly never used it myself.

> 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.


Where does ZF say anything like that?

WM is far to ignorant of what ZF actually does say to be able prove that
ZF says anything like that.


> Just this causes the contradiction. A union of finite initial
> segments cannot have a number of elements that is larger than every
> finite number.


The union of infinitely many finite initial segments, each
size-comparable with all the others and no two of the same size can
certainly not be of any finite (non-infinite) size.

At least not outside WMytheology.
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