how to list all of the real numbers

All the reals between zero and one, that is. There's nothing
new here, but with all the recent talk about Cantor's diagonal
proof revealing the impossibility of making one kind of
list of the reals, this kind of list might bear repeating:

Consider the reals between zero and one in base 2, so
that we can look at all possible endless sequences
of zeros and ones as the binary tree emanating from a
single zero, before the binary point, with all else following
the binary point, and spreading downward like a real tree.

We can make it a perfectly and completely imagined
list of all of these reals by having each node branch to
a zero node on the left and a one node on the right. Thus
each endless path corresponds to a real number, and every
real number is represented by a path. Some of the reals
are represented by more than one path, but that doesn't
matter for our purpose, which is to *explicitly* list all of the
real numbers between zero and one.

As is well known, the number of paths is uncountable,
though the number of nodes is countable.

Thus, amazingly enough, we can list ALL of these reals
at once, to any desired binary place, simply by working
on the tree, from left to right, row below row of nodes,
for a finite number of steps. And we can calculate that
number of steps.

'All at once' is misleading, of course, since multiple
reals occupy the same paths to whatever desired binary
place we choose to stop. But they're all there.

Could there be a more clear and complete and intuitive
way of illustrating, not only the real numbers, but also
an uncountable set? The power of the continuum is
right here before our eyes.

.

Relevant Pages

  • Re: Math, The Friend of Relativity
    ... Don't be reassured by the ranting of Uncle Bonehead, ... As if we would believe something just because Einstein said so. ... zero, either, by the same argument Uncle Ben gave for real numbers. ... I've already proven h does NOT belong to the set of reals, ...
    (sci.physics.relativity)
  • Re: in fact zero _is_ neither positive nor negative. THIS STATEMENT IS NOT TRUE
    ... So in fact zero can be negative and can be positive and it just ... which most then build the reals from. ... and agree that magnitude is much more fundamental ... and if you want to use the bourbaki "positive" (positif ?) to include ...
    (sci.math)
  • Re: The common usage of "nonnegative real number" is ludicrous.
    ... not necessarily zero, and that is usually what is meant by ... a complicated construction. ... When I want to talk about magnitude I am talking about ... reals contain zero or not is just convention. ...
    (sci.math)
  • Re: the binary tree
    ... Please demonstrate the first several layers of this binary tree. ... another zero and one. ... Then you agree that you are constructing a subset of the rationals ... that is dense in the reals. ...
    (sci.math)
  • Re: On writing negative zero - with or without sign
    ... exact zero, contrary to your claim that there is no such thing. ... that using float for something *other* than simulating continuous reals ... "exactitude" in general, but rather a very specific issue, namely ...
    (comp.lang.fortran)