Date: Dec 22, 2009 9:28 AM
Author: Dik T. Winter
Subject: Re: Another AC anomaly?
In article <firstname.lastname@example.org> WM <email@example.com> 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.
> 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
> > 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
> 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
Yes. But as you have stated that your tree contained finite paths only,
such an infinite initial segment is not (according to *your* definition)
> 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/