Date: Jan 23, 2013 2:15 PM
Author: Virgil
Subject: Re: ZFC and God

In article 
WM <> wrote:

> On 23 Jan., 12:47, "Jesse F. Hughes" <> wrote:
> > WM <> writes:
> > > On 22 Jan., 21:18, "Jesse F. Hughes" <> wrote:
> > >> WM <> writes:
> >
> > >> > That is potential infinity. That proof is not necessary, because the
> > >> > set is obviously potentially infinite. No, you shoudl give a proof,
> > >> > that there is a larger k than all finite k.

> >
> > >> Er, no.  When I say that the union is infinite, I do not mean that it
> > >> contains an infinite number.

> >
> > > But you mean that the tree contains infinite paths. And just that is
> > > impossible without ...

> >
> > > In order to shorten this discussion please have a look at
> >
> >
> >
> > No.  It's irrelevant.

> You are in error.

The moderators of that URL did not think so, since they deleted the
referenced posting.
> >
> > We're talking about whether you can prove that
> >
> >  U_n=1^oo {1,...,n}
> >
> > is finite.  I'm not switching topics to paths in trees (despite the
> > fact that the ignorance of your question is obvious).

> The union of FISs is finite. Yes that is my claim. But I cannot give
> an upper limit, because the finite numbers have no upper limit. This
> is called potentially infinite.

Being potentially infinite is like being "a little bit pregnant". In the
real world and in the mathematical world, pregnancy and infiniteness are
on and off characteristics, with no middle ground between being and not
> >
> > > There it has meanwhile turned out ... But see it with your own eyes
> > > what you would not believe if I told you.

> >
> > > The index omega is in reach, it seems.
> >
> > You're playing your usual little game of trying to change the topic.
> > I won't have it.
> >
> > I take it that this new tack is so that you don't have to concede the
> > point: there is no mathematical publication which claims that the
> > above union contains elements larger than any natural, nor any
> > publication which claims that this is what it means to be infinite.

> I know. But if you hace read the discussion, you have seen that two
> matheologians claim just this.

Actually whether |N contains elements larger than ANY natural is a bit
ambiguous, and WM relies on such ambiguity to confuse the issue.

> 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?

Finite binary trees are as much unions of all their levels as any
Complete Infinite Binary Tree.

And in any such complete tree any set of nodes which contains the root
node and contains one and only one child node of each of the sets parent
nodes is a path in that tree.

> Are there only the
> finite paths?

Only in finite trees.

> Or are there also the infinite paths?

Only in infinite trees.

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

WM may need such a level but no one else does.

In a Complete Infinite Binary Tree the set of "levels" bijects with |N
which also does NOT contain an "omegath" member.