Re: Cantor and the binary tree



Martin Shobe wrote:
> On 11 Jun 2005 06:29:14 -0700, mueckenh@xxxxxxxxxxxxxxxxx wrote:
>
> >
> >
> >Randy Poe wrote:
> >> mueckenh@xxxxxxxxxxxxxxxxx wrote:
> >> > Is here a bijection possible between uncountable sets?
> >>
> >> Sometimes.
> >>
> >> There exist bijections between R and C, for instance. But
> >> not between R and P(R).
> >>
> >> > Can you enumerate R <--> R?
> >>
> >> I can construct a bijection from R to R, if that is what
> >> you are asking.
> >>
> >> Here's one: f(x) = x.
> >
> >And I can construct a bijection from nodes to paths:
> >
> > 0.
> > / \
> > 0 1
> > /\ /\
> > 0 1 0 1
> > /\ /\ /\ /\
> > ...................
> >
> >Everywhere a path branches off, it gets the node. The other one keeps
> >that node which already was mapped on it before. I need not knwow which
> >path gets which node, because all are of similar value. And so on in
> >infinity.
>
> That is not a bijection.
>
> If you compare, '000...' with '010...', you would map '000...' to '0'
> and '010...' to '01'. If you compare '000...' with '011...', you
> would map '000...' to '0' and '011...' to '01'. QED.

There is a path P (for instance 1/3 = 0.010101... = P) This path has
got a node (a node is mapped on it). In the same edges there are many
other paths, but that does not bother us. They will get their nodes
later on. From the path P and its companions many other paths separate
in the next node. Among them is the path P'. The node, where this
happens, is mapped on P'. Its companions remain nodeless nude.
So we have two nodes mapped on two paths, P and P'. The next time when
a bunch of paths separates from P, *one* of those gets the node where
that happens. (Same for separation of a bunch from P' and that bunch
remaining with P'.) The others do not yet get nodes but will have to
wait until their turn will have come. So every path is either equipped
with a node or it has not yet been separaed from other paths or it has
alreay its node but is not yet separated from all others. In any case
we make sure that an individual path cannot exist separated from all
others without carrying a node. Whether individual paths separated from
all others do exist at all, that is another question.

There is no loophole: An individual path separated from all the others
cannot exist without its node. This means in translation: If real
numbers do exist as individual entities then they form a countable set.

Regards, WM

.