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

rgheck rgheck at brown.edu
Sat Oct 11 14:15:26 EDT 2008

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?
Yes, of course, it does imply this, assuming the theory in question is 
recursively axiomatized. But the incompleteness in question will be 
mathematical in nature, not physical. So it does not follow immediately 
that there are any (interesting) physical facts that the theory does not 
entail. Of course, there may be some physical facts it does not entail, 
if it's possible to code enough arithmetic in the purely physical part 
of the theory. But that's why I included the word "interesting" in 
parentheses. These very well may not be physical facts of any interest.

Here's another way to look at this. Our overall recursively axiomatized 
physical theory T contains as a part some mathematical theory M, stated 
in some language L. (Maybe M is analysis and L is the language thereof.) 
Now consider the theory T+ in which M is replaced by M+, the set of all 
truths of L. The interesting question for a physicist, I take it, would 
be whether T+ was still incomplete. The *mathematical* incompleteness 
isn't so important.

Richard Heck

Richard G Heck Jr
Professor of Philosophy
Brown University

More information about the FOM mailing list