[FOM] Completeness formulations

[joeshipman@aol.com]

The place Godel Incompleteness can matter in physics is if the 
consequences of a mathematized physical theory are mathematically 
*definable* without being mathematically *computable*.  For example, if 
a TOE involved a physically measurable dimensionless quantity which was 
a nonrecursive real number, then physics could allow the derivation of 
new mathematical facts that were beyond the capacity of ZFC to prove. 
(ZFC or any recursively axiomatized theory can only settle the value of 
finitely many bits of precision for any definable nonrecursive real 
number before reaching a bit whose value is an unprovable mathematical 
fact which may nonetheless be measurable to a high degree of confidence 
if you believe the "TOE".)

Conversely, Godel Incompleteness would mean that the results of the TOE 
could not be predicted with mathematical certainty to arbitrary 
precision.

This is not as farfetched as it might at first appear. For example, 
Hartle and Hawking had a theory of quantum gravity in which physically 
measurable quantities were defined in terms of a sum over equivalence 
classes of 4-dimensional simplicial topologies, where the unsolvability 
of the homeomorphism problem means that no method of calculation was 
apparent. That doesn't prove that some clever way could not be found to 
perform the summation without enumerating the distinct cases, but it's 
conceivable that a suitable theory could involve such a summation 
having an r.e. complete real number as its answer. (This is discussed 
in my 1992 paper "Aspects of Computability in Physics".)

-- JS




On Oct 9, 2008, at 10:43 PM, Brian Hart wrote:

>
> 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?  Shouldn't the constructors of the
> Theory of Everything be alarmed?  I know this suggestion of
> application of Godel's theorem was made decades ago but why didn't it
> make a bigger impact?  Is it because it is wrong or were there some
> sociological reasons for mainstream ignorance of it?
> _