Re: Another set with cardinality |Z|

From: Ralph Hartley (hartley_at_aic.nrl.navy.mil)
Date: 09/27/04 

Date: Mon, 27 Sep 2004 09:40:34 -0400

Eray Ozkural exa wrote:

> Hmm. In binary, it will go like
>
> 0.01010101...
>
> You mean that it cannot generate because this rational number has an
> infinite binary expansion. I am not sure if that is an intuitively
> acceptable explanation, though, simply because of the fact that the
> tree has an infinite number of nodes, e.g. an infinite depth,
> therefore this number is computed in the limit.

Ah! "in the limit"

Lots of things happen when you look at limits.

Infinite trees don't have leaf nodes. But if you wanted to talk about
something *like* leaf nodes, you could add "limit nodes" each of which is
defined as the limit (relative to some metric) of a path down the tree.

Not all trees have such limits, but yours does, and they *are* the reals
(in [0,1]).

Your procedure is essentially equivalent to (one of the) traditional way(s)
of defining reals as the limit of sequences of rationals.

What Cantor proved is that the number of limit nodes is bigger than the
number of nodes in your original tree.

It might be OK to think of your procedure as "describing" the set of all
reals, but it would be a complete redefinition of the word to think of it
as "computing" all reals.

After all, there are more limits of sequences of computations than there
are computations. (proof (perhaps not the simplest possible): There are
countably many programs. All rationals are computable, so any Cauchy
sequence of rationals is produced by a sequence of programs, so any real is
the limit of the output of a sequence of programs, so there are uncountably
many different sequences of programs.)

Ralph Hartley

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