[FOM] Completeness formulations

[Harvey Friedman]

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?
> _

This question suggests the following formulation.

To focus matters, let's assume that we are talking about a theory  
where the mathematics involved is put in purely set theoretic terms.  
Of course, there are modifications of the setup below that may be  
preferable for some purposes.

So we are looking at first order theories T in many sorted predicate  
calculus with a sort for sets, with epsilon between sets, and equality  
between sets.

We have the following concepts.

COMPLETENESS_1. Any two model of T in which the sort for sets is  
interpreted as "all sets, with ordinary membership and equality"  
satisfy the same sentences.

COMPLETENESS_2. Any two models of T in which the sort for sets is  
interpreted as "all sets, with ordinary membership and equality" are  
isomorphic.

We can also allow second and higher order theories T, with the usual  
second and higher order semantics.

Now these formulations are obviously made in class theory. If we want  
formulations in set theory, then we would not use the entire set  
theoretic universe for the set theoretic sort, but instead some  
fragment thereof, renaming the sort as the such and such restricted  
set sort. E.g., the set sort might be "the hereditarily finite set  
sort", or "the rank < omega + omega set sort".

Here is a semi formal conjecture:

CONJECTURE. For any reasonably system that purports to formalize some  
physical theory incorporating mathematics, if it is Complete_1 then it  
is Complete_2. Furthermore, if it is not Complete_1, then there is a  
simple sentence that violates Complete_1.

In particular, this Conjecture suggests that if we look at formalized  
physical theories, and we factor out the math incompleteness, then we  
get completeness issues that are of a totally different character than  
those in mathematics.

Harvey Friedman