On 22-Jun-01, [EMAIL PROTECTED] 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
