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/