Date: Dec 22, 2009 9:28 AM
Author: Dik T. Winter
Subject: Re: Another AC anomaly?

In article <65fa0fa6-3723-4b81-ad46-1c3ab274fccc@t42g2000yqd.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 21 Dez., 14:17, "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?

>
> Except in matheology it is always required.


I did not know of that requirement. Can you provide for a reference where
that requirement is mentioned?

> > > 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.

>
> That's why you love matheology.


I would have thought that you would be able to provide for a textbook where
that requirement is mentioned. So give me one.

> > > > > 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.

>
> It may create what you like. Either 1/3 can be identified at a finite
> digit or 1/3 cannot be identified at a finite digit.


Not.

> Even a matheologian should understand that: If there is no digit at a
> finite place up to that the sequence 0.333... identifies the number
> 1/3, then there is no digit at a finite place up to that the number
> 1/3 can be identified.


Right, there is no digit at a finite place up to that the number 1/3 can be
identified. And as there are no digits at infinite places that appears to
you to be a paradox. It is not. There is *no* finite sequence of digits
that identifies 1/3. But there is an *infinite* sequence of digits that
does so.

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

> >
> > which you contested.

>
> I did not contest it.


Why then did you reply with:
> That is pure opinion, believd by the holy bible (Dominus regnabit in
> aeternum et ultra. [2. Buch Moses: Exodus 15 Vers 18]) or forced upon
> us by the men-made axiom of infinity.

it that is not contesting it?

> I said, if there is a sequence that identifies
> 1/3, then the identifying digits must be at finite places.


Right, all identifying digits (there are infinitely many) are at finite
places.

> But we know
> that for every finite place d_n, there is a sequence d_1, ..., d_n
> that is not 1/3 but is identical to the sequence of 1/3. Therefore we
> can conclude that there is no sequence identifying the number 1/3 by
> means of digits at finite places only.


You can only conclude that there is no *finite* sequence that identifies
the number 1/3. You here exclude the possibility of an infinite sequence
of digits at finite places only, i.e. assuming that what you want to prove.

> > > > 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.

>
> The union of finite initial segments cannot ield an infinite initial
> segment?


Yes. But as you have stated that your tree contained finite paths only,
such an infinite initial segment is not (according to *your* definition)
a path.

> Does the sequence of 1/3 not consist of a union of all finite
> initial segments?


It is, but also (according to *your* definition) it is not 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/