Re: Countable vs Continuous

From: Brent Meeker <>
Date: Fri, 22 Jun 2001 10:46:16 -0700

On 22-Jun-01, 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
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

This archive was generated by hypermail 2.3.0 : Fri Feb 16 2018 - 13:20:07 PST