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/