FOM: RE: 157:Finite societies
Matt Insall
montez at fidnet.com
Wed Aug 14 10:33:36 EDT 2002
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
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