Date: Dec 21, 2009 8:17 AM
Author: Dik T. Winter
Subject: Re: Another AC anomaly?

In article <2fac8bb1-4c90-4421-b559-1ea7f0301d4f@e27g2000yqd.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 18 Dez., 15:12, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > Why need I to think about a last one (which there isn't) to be able to
> > think about a set that contains all natural numbers? Apparently you
> > have some knowledge about how my mind works that I do not have.

>
> Yes. A very convincing and often required proof of completenes of a
> linear set is to know the last element.


Oh, is it often required?

> T talk about all in case there
> is no last is silly.


And I think it is silly to require there being a last to be able to talk
about all.

> > > > > > Right, but there is no finite initial segment that contains them
> > > > > > all.

...
> > > > Sorry, I have no knowledge of the bible. But live without that axiom
> > > > when you can't stomach it. And do not attack mathematicians who
> > > > live with that axiom.

> > >
> > > To live with that axiom does not create uncountability. See the proof
> > > here:
> > >http://groups.google.com/group/sci.logic/browse_frm/thread/46fa18c8bb=

> ...
> >
> > Where is the proof there? I see only you writing a bit of nonsense and
> > two rebuttals.

>
> One of the rebuttals has meanwhile been changed. Peter Webb
> recognized: It is true that you cannot show pi as a finite decimal,
> but you can't show 1/3 as a finite decimal either.


So what? That is not contested and it does not show in *any* way that the
axiom of infinity does not create uncountability. So no proof at all.

> Just what I said.

And just wat I said: see the quote above:
> > > > > > Right, but there is no finite initial segment that contains them
> > > > > > all.


which you contested.

> > > > The infinite paths because you stated a priori that your tree did
> > > > not contain infinite paths. So it is impossible to construct in
> > > > your tree infinite paths by the axiom of infinity.

> > >
> > > The axiom of infinity establishes the set N from finite numbers.

> >
> > It establishes the *existence* of a set N of finite numbers.

>
> What else should be established?


Does not matter. The axiom of infinity does *not* construct infinite paths
in your tree, beacuse you stated that your tree did not contain infinite
paths a priori. So those infinite things are not paths by your statement.
Neither does the axiom of infinity establish a finite set N of all finite
numbers.

> > > It establishes the infinite paths as well in my tree from finite
> > > paths.

> >
> > No. That is impossible because you stated that the paths were finite.
> > What it *does* establish is the extistence of a set P of finite paths.

>
> It is rather silly to argue about the uncountability of the set of
> paths. Only minds completely disformed by set theory could try to
> defend the obviously false position that there were uncountably many
> paths.


But: if you consider only finite sequences of nodes as paths, there *are*
countably many paths. You continuously confuse what you consider being
a path and what others consider a path.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/