FOM: RE: 157:Finite societies

[Matt Insall]

Hello Harvey,
     I truly enjoyed reading about finite societies.  Thank you for these
interesting posts.  I have a question about your theorem number 3, which I
shall recall for reference here:

``THEOREM 3. For any k,m, there is a weak finite society (S,L,W,A,k) with
exactly m ages satisfying condition alpha. This statement is provably
equivalent to the consistency of ZC + {there exists V(omega times n}_n,
over PRA. In particular, the statement is provable in ZC + (forall
n)(V(omega times n) exists) but not in ZC + {V(omega times n) exists}_n.''.

Have you characterized those fragments T of ZFC for which such a result
holds?  That is, for which theories T\subseteq ZFC can we prove the
following?

For any k,m, there is a weak finite society (S,L,W,A,k) with
exactly m ages satisfying condition alpha. This statement is provably
equivalent to the consistency of T + {there exists V(omega times n}_n, over
PRA. In particular, the statement is provable in T + (forall n)(V(omega
times n) exists) but not in T + {V(omega times n) exists}_n.


Is ZC the ``best'' fragment of ZFC for which this is the case?  Also, is
there a specific property P of fragments of ZFC that makes the proof go
through, or for which you can prove a tehorem like the following?

A theory T\subseteq ZFC has property P if and only if the statement ``For
any k,m, there is a weak finite society (S,L,W,A,k) with exactly m ages
satisfying condition alpha.'' is provably equivalent to the consistency of T
+ {there exists V(omega times n}_n, over PRA.



Matt Insall