Re: the binary tree

On Apr 17, 1:11 pm, Dave <dave_and_da...@xxxxxxxx> wrote:
On Apr 17, 11:29 am, calvin <cri...@xxxxxxxxxxxxxx> wrote:
On Apr 17, 11:59 am, Dave <dave_and_da...@xxxxxxxx> wrote:
On Apr 17, 10:41 am, calvin <cri...@xxxxxxxxxxxxxx> wrote:
On Apr 17, 11:29 am, Dave <dave_and_da...@xxxxxxxx> wrote:
On Apr 17, 8:23 am, calvin <cri...@xxxxxxxxxxxxxx> wrote:
Yes, but so what?  The binary tree yields the uncountable
number of reals between zero and one in a countable
number of construction steps.
Please demonstrate the first several layers of this binary tree.
It's just the standard binary tree:

Under each node, consider the left branch to be a zero
and the right branch to be a one.  At the end of each
branch is a node, from which drop two more branches,
another zero and one.  Under the topmost node, where
I place the binary point, is the first row, consisting
of two branches ending in two nodes.  Under that is the
second row, consisting of four branches and four nodes;
under that is the third row, consisting of eight branches
and eight nodes.  And so on.

Are there values in the nodes? If so, what are they? If not, where do
all the real numbers lie?

The values are in the branches, the way I visualize it,
but I suppose it could be visualized as the nodes at
the end of the branches having the values.

Let's say that such a tree has just two rows.
Then the four values represented are .00, .01,
.10, .11 ; the same result whether the branches
or the nodes are assumed to have values.

It's not that the real numbers 'lie' anywhere, though
the tree can be constructed as I noted in a previous
post, where each path approaches a real number on the
unit line interval at the bottom of the diagram.  But
the real numbers are represented by the possible
paths, of which there are an uncountable number.
Two such paths are .000... and .111... with the
represented values being zero and one.  An example
of an irrational number represented by a path is
.01001000100001000001...  Please let's not quibble
about what '...' means in these examples, not that
you were about to do so.

Then you agree that you are constructing a subset of the rationals
that is dense in the reals.

I was thinking that 'dense' might have a more subtle
definition than it turns out to have:

http://www.abstractmath.org/MM/MMRealDensity.htm

I'm quite familiar with the meaning of dense at the
link above. As for your question, no, I don't agree that
I was constructing a subset of the rationals that is dense
in the reals. I was just constructing (and of course
it was not my invention) the complete binary tree in
a countable number of steps. The tree contains all
binary representations of real numbers (plus repititions)
between zero and one, assuming a binary point at the top
of the tree.
.

Relevant Pages

  • Re: the binary tree
    ... number of reals between zero and one in a countable ... Please demonstrate the first several layers of this binary tree. ... another zero and one. ...
    (sci.math)
  • Re: the binary tree
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    (sci.math)
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  • Re: how to list all of the real numbers
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