[FOM] Godel's First Incompleteness Theorem as it possibly relatesto Physics

[hendrik@topoi.pooq.com]

On Mon, Oct 13, 2008 at 01:16:38PM -0400, Brian Hart wrote:
> On Sat, Oct 11, 2008 at 12:36 AM, Kreinovich, Vladik <vladik at utep.edu> wrote:
> >
> >
> >> -----Original Message-----
> >> From: Brian Hart
> >
> >> Why doesn't Godel's 1st Incompleteness Theorem imply the
> >> incompleteness of any theory of physics T, assuming that T is
> >> consistent and uses arithmetic?
> >
> > It does.
> >
> >> Shouldn't the constructors of the
> >> Theory of Everything be alarmed?
> >
> > The designers of the Theory of Everything are interested in predicting
> > all possible events -- i.e., in effect, all possible ATOMIC statements.
> > Goedel's theorem shows that one cannot easily predict the truth values
> > of the corresponding quantifier statements, but this is not of any
> > serious concern to physicists.
> >
> > If a physical theory like Newton's gravity can predict the position of
> > all the planets in any given future moment of time, physicists will be
> > very very happy.
> 
> I don't think it can, though.  For example, isn't the n-body problem
> within Newtonian gravity where n >= 3 intractable?

Only intractable in the sense that there isn't a neat formula 
that solves it -- there's just a differential equation.  Nothing 
prevents the numerical analyst from computing arbitrarily close 
approximations except time and effort.  Isn't that more or less the 
definition of computable reals?

> One of the first
> observable improvements upon Newton's formulation of the theory of
> gravity made by General Relativity, for example, was to correctly
> account for the precession of the perihelion of Mercury.

And that has nothing to do with the computational feasibility of either 
theory.

-- hendrik