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Re: Another AC anomaly?
Posted:
Dec 18, 2009 9:12 AM
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In article <a737fea3-6dd2-4a1d-9506-2df7cb6d412a@v30g2000yqm.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 17 Dez., 14:44, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> .com> WM <mueck...@rz.fh-augsburg.de> writes:
> > > On 16 Dez., 03:31, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > > > Why that? Group, ring and field are treated in my lessons.
> > > >
> > > > You think that something that satisfies the ZF axioms being a
> > > > collection of sets is rubbish, while something that satisfies
> > > > the ring axiomsbeing a ring is not rubbish?
> > >
> > > Yes, exactly that is true.
> >
> > And why, except by opinion?
>
> Look here:
> http://groups.google.com/group/sci.logic/browse_frm/thread/46fa18c8bb5da79d#
What is the relation?
> > Aha, you are clearly a mindreader. Well, as far as I know mindreading
> > is not part of mathematics. Anyhow, I can think of numbers larger than
> > that path.
> > But that is completely irrelevant. I am able to think about a set that
> > contains all natural numbers, you apparently are not.
>
> How do you know that without confirming it by thinking the last too?
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.
> > > > Right, but there is no finite initial segment that contains them all.
> > >
> > > 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.
> >
> > 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/46fa18c8bb5da79d#
Where is the proof there? I see only you writing a bit of nonsense and two
rebuttals.
> > > The tree contains all paths that can be constructed by nodes, using
> > > the axiom of infinity. Which one would be missing?
> >
> > 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.
> 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.
> > You can construct infinite
> > sequences of nodes, but as you stated *explicitly* that your tree did
> > not contain infinite paths, those infinite sequences of nodes are
> > apparently not paths within your terminology.
>
> They are not constructed, but it might happen that the come in by the
> union of some finite paths: The sequence of finite paths 1, 11,
> 111, ... may yield an infinite path 111... as a limit.
How is it possible to yield an infinite path if by your definition paths
are finite? That is like saying lim(n -> oo) is a natural number.
> > > > In that case you have a very strange notion of "existing in the tree".
> > > > Apparently you do *not* mean "existing as a path". So when you say
> > > > that the number of (finite) paths is countable, I agree, but 1/3 is
> > > > not included in that, because it is not a path according to your
> > > > statements.
> > >
> > > It is. I constructed a finite path from the root node to each other
> > > node.
> >
> > Yup, you constructed a finite path, and that does not represent 1/3.
>
> It 1/3 can be represented by a bit sequence, then it is represented in
> my tree by the union of the node sets representing
> 0.0, 0.01, 0.010, ...
Right, but that is not a path by your definition.
> >
> > > Then I appended an infinite tail.
> >
> > Whatever that may be, it is *not* a path according to your explicit
> > statement that the tree did not contain infinite paths.
>
> You have misread my construction. I use finite paths and then I append
> an infinite tail.
But that is not a path by your statement.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
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