On 22-Jun-01, juergen.domain.name.hidden wrote:
>> or continous. Don't the computable numbers form a continuum; hence
>> even restricting the universe to one we can describe would still
>> allow it to be continuous?
>>
>> Brent Meeker
>
> No, the computable numbers do not form a continuum - there are not
> more than countably many of them. Any real number computable in the
> limit (such as Pi) has a finite nonhalting program; the set of all
> such programs cannot have higher cardinality than the integers.
>
> Juergen Schmidhuber
>
> http://www.idsia.ch/~juergen/
> http://www.idsia.ch/~juergen/everything/html.html
> http://www.idsia.ch/~juergen/toesv2/
>
Thanks for the reply, Juergen. I guess I didn't phrase my question
right. I know that the cardinality of the computable numbers is the
same as the integers. What I was asking was whether the computable
numbers form a continuum in the topological sense (I'm pretty sure they
do) - AND - is this a sufficient continuum to provide a model of
continuous space-time? Again, I think it is - but I don't know of a
proof one way or the other.
Brent Meeker
Received on Fri Jun 22 2001 - 11:52:08 PDT