Re: Cantor and the binary tree



Virgil wrote:
> In article <1118495925.082216.67690@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> mueckenh@xxxxxxxxxxxxxxxxx wrote:
>
> > Virgil wrote:
> >
> > > How does one determine, in a maximal binary tree, which two nodes get
> > > surjected onto which maximal path? Your allegation gives no way of
> > > deciding that question, so any such alleged surjection so far exists
> > > only in your mind.
> > >
> > > Can you make it real?
> >
> > Yes.
> >
> > 0.
> > / \
> > 0 1
> > /\ /\
> > 0 1 0 1
> > /\ /\ /\ /\
> > ...................
> >
> > I count the nodes: 1,2,3,... and map each of them on a path.
> >
> > 1
> > / \
> > 2 3
> > /\ /\
> > 4 5 6 7
> > /\ /\ /\ /\
> > ...................
> >
> > Node 1 (=0.) is mapped on 0.111...
> > Node 2 is mapped on 0.010101... = 1/3
> > Node 3 is mapped on 0.1010111...
> > Node 4 is mapped on 0.00111000111100001111100000...
> > And so on. All the other paths are related with other nodes, because I
> > know that there is one node for each path which separates itself from
> > those already carrying nodes.
>
> First: This does not unambiguously define any rule, as there are
> uncountalby many assignment schemes that agree through the 4th node and
> disagree thereafter.

There are many schemes to count he rational numbers. One of them is
sufficient to prove the countability.
>
> Second: Paths and binary place value expansions are not in 1 to 1
> correspondence , at least in the obvious way since, for example, 0.1(0)
> and 0.0(1), where the parenthesized string of digits repreats forever,
> represent different paths but the same real number.

The nodes count all binary representations. So they count even a bit
more than all reals of (0,1). Is that a disadvantage in your eyes?
>
> So I will omit the "0." in representing a path by a string.
>
> Which node is mapped onto the path 0110001111000001111110000000..., one
> 0 followed by two 1's followed by three 0's followed by four 1's, etc.?

I have deviced a list, which is admissible because the nodes are
counatble. To answer your question, node 197776657875786899 is maped on
the number in question.
>
> There are, in fact, uncountably many such possible strings for which no
> node can be assigned by any rule.

You see by my tree that no path can separate itself from another
without a node. Of course, always many separae from many others, but
the node is mapped on one of them. The others will get their nodes
later, according to my countable list.
> >
> > >
> > > And there is no such thing so far defined in any tree as a "partial"
> > > node, so absent such a definition, you are only blowing smoke.
> >
> > It would be easy to understand for a mathematician. But you don't
> > want to or cannot.
>
> Mathematicians are notorious for requiring precise definitions before
> allowing such ambiguous phrases as "partial nodes" to be used. So I am
> just being mathematician-like in refusing to accept such hand waving
> without proper definition of the thing being waved.

Mathematicians are notorious of understanding complicated thnigs quite
easy. That partial node is a very simple constructuion. But wI do no
longer use it in order to keep things understandable for you.

Regards, WM

.

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