Re: abundance of irrationals!)

Virgil said:
> In article <1115808674.466909.326260@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> mueckenh@xxxxxxxxxxxxxxxxx wrote:
>
> > Russell wrote:
> >
> > > No. The obvious linear enumeration of *nodes* is not an
> > > enumeration of the reals. For that, you need to enumerate
> > > the different paths. That's not merely a rearrangement of
> > > the enumeration of nodes; if you think it is, then please
> > > tell us how you associate each node with one and only one
> > > path. Can't be done.
> >
> > Tell us how you associate Cantor's antidiagonal with one and only one
> > line, e.g., where it ends. Cant't be done.
> >
> > But the nodes can be shown to be not less than the paths.
>
> I challenge you to show this. I claim that one can ennumerate the nodes
> but not the paths.
>
> Each node can be represented by a finite string of zeros and ones, the
> root node by one digit string, "1", and given any finite string, "s",
> representing a node, the left and right nodes at the next level by "s0"
> and "s1", respectively.
>
> Since interpreting these as binary integers counts them, they are
> countable.
>
> For the infinite strings needed to represent paths, a somple diagonal
> costruction will show that no list can be complete.
>
>
>
> > It is not
> > possible to say which node belongs to some single path.
>
> Why not?
>
There is a 1-1 correspondence between the points and lines, the nodes and
branches. You have not specified exactly what you mean by branch, so i will
define it as a line connecting two nodes. Perhaps you mean a string of such
connections, in which case that is also specified by the string of digits
denoting each number, 0 for left branch and 1 for right.

We start with the root node. It has no corresponding branch, but you could hang
it from one if you wanted, perhaps off the null set of digits. No matter. Every
additional node you add requires precisely one additional branch segment, to
attach it to its parent node. This is obvious.
--
Smiles,

Tony
.

Relevant Pages

  • Confused by implementing an IEnumerable with iterator blocks
    ... I wanted to write a class to encapsulate an enumeration of all the files in a directory. ... but when hundreds of files are involved and the directory is on a remote server this causes a substantial delay, so I wanted to design an enumeration that would browse through the root directory and its subdirectories incrementally as iteration took place. ... having read up only briefly on how "yield" works: ... public string Root ...
    (microsoft.public.dotnet.languages.csharp)
  • Re: Question regarding the diagonal method.
    ... denumerable binary strings, a string 111... ... the enumeration. ... Then define Sby looking at blocks of three digits at a time; ...
    (sci.logic)
  • Re: enum.ToString()
    ... public static string GetEnumValueDescription ... DescriptionAttribute pobjAttribute = ... How do I display the descriptive string ... you should use attributes to attribute the>> elements in the enumeration. ...
    (microsoft.public.dotnet.languages.csharp)
  • Re: Generic List Find.
    ... Elasped time for Enumeration is 00:00:13.0926257 ... Dim Products As New List ... Private Function FindProduct(ByVal code As String) As Product ...
    (microsoft.public.dotnet.languages.vb)
  • Re: Generic way of writing this enum code?
    ... // Check to make sure that T is an enumeration. ... // Create the string builder. ... because there's no constraint making T convertible ... foreach (EnumType1 type in types) ...
    (microsoft.public.dotnet.languages.csharp)