In article <email@example.com> WM <firstname.lastname@example.org> 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 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/