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