Date: Jan 23, 2013 11:02 AM
Subject: Re: ZFC and God

On 23 Jan., 14:36, "Jesse F. Hughes" <> wrote:
> WM <> 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