🚀 go-pugleaf

In article <1992Nov18.133219.6664@sei.cmu.edu>, firth@sei.cmu.edu (Robert Firth) writes:
> In article <1992Nov17.124233.24312@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:
> 
> Necessary, but not, I fear, sufficient.  If you are to replace the
> continuum with a *single* discrete manifold, then you must show that,
> for any uncountable set with certain properties, there exists a
> *single* countable set with all the same properties.

I do not see how these two are related. A fully discrete model
will either be consistent with experimental results or not. Its validity
has nothing to do with questions about countability. A discrete model
will be completely finite, unless our universe is potentially infinite.
Even if it is potentially infinite, any part of the universe we know 
at a given time will be finite.

> I freely grant that we still have trouble defining a single continuous
> manifold that accommodates all our favourite theories (like GR and QM),
> but I can't see how dropping the assumption of continuity will make
> this problem any easier to solve.

In a fully discrete model there are no true singularities. What appears
to be moving toward a singularity when viewed at a macroscopic level
always resolves itself in some other way. Since most scientist suspect
this is what happens physically, this is an argument for thinking
that such models might make sense physically.

I think it will be difficult to build a discrete theory that
encompasses GR and QM. The issue is not how difficult the theory is, but
what approach gives the simplest model that accounts for experimental
results. Arguments about countable sets have little relevance to this
question. They do allow one to avoid making certain errors such as
thinking that the continuum is a necessary mathematical element in
any physical theory.

Paul Budnik