Bruno wrote:
>Juergen Schmidhuber:
>> Even the set of all real #'s is not formally describable.
You cannot
>> write a program that lists all reals. In infinite but
countable time
>> you can write down a particular computable real, but not
all reals.
>I don't agree. Indeed, you cannot write a program that
lists all reals.
>But in infinite and still countable time you can write down
all the reals,
>by dovetailing on all initial segment of all the reals,
including
>non computable one.
> 0.0 0.1
> 0.00 0.01 0.10 0.11
> 0.000 0.001 0.010 0.011 0.100 0.101 0.110
0.111
> etc. etc.
>There is of course no contradiction with the fact that most
of the reals
>are uncomputable, and that the set of reals is uncountable.
No you've got that wrong, Bruno. The output of the process you describe is a
countable string of reals, and it thus leaves out the vast majority of all
real numbers. (To see that the string is countable, map it to the integers
as follows: upper-left=1, upper-right = 2, second row-first column = 3 etc.;
this mapping will include all entries in your listing) -- What you are
leaving out are a lot of reals that have an infinite binary decimal
representations.
Nick Bostrom
Dept. Philosophy, Logic and Scientific Method
London School of Economics
Email: n.bostrom.domain.name.hidden <mailto:n.bostrom.domain.name.hidden.ac.uk>
Homepage:
http://www.analytic.org <
http://www.analytic.org>
Received on Tue Nov 02 1999 - 05:44:48 PST