[FOM] Incompleteness and Physics

[Vaughan Pratt]

joeshipman at aol.com wrote:
> On the other hand, Godel Incompleteness also implies an influence of 
> physics on mathematics -- as Godel pointed out, we might come to 
> believe in new mathematical axioms because they made physics work. This 
> will be the case if, according to some mathematized physical theory, 
> there exists a definable but noncomputable real number that is 
> measurable within the theory -- ZFC-proofs could only settle the value 
> of finitely many bits of such a number, possibly fewer than can be 
> measured.

Given that every finite set of natural numbers is representable in ZFC, 
no matter how precisely one measures a given quantity there will always 
be a ZFC sentence that can "account" for it.

If there is a notion of "physical theory" that rules out certain finite 
subsets of N that as dyadic rationals can't be an approximation (to that 
accuracy) of any fundamental constant of any physical theory, that would 
be extremely interesting.  If not, I have difficulty understanding what 
it would mean to have a finite measurement outside the scope of all 
ZFC-based theories.

Vaughan Pratt