[FOM] S4 + ZFC
Michael Carroll
mcarroll at pobox.com
Fri Aug 24 15:24:19 EDT 2007
Rupert McCallum wrote:
"Surely conservativity is obvious. Just get rid of all the boxes."
Conservativity clarifies what's at issue. In general a conservative
extension expands the set of theorems but restricts the set of models. The
proofs that AC is independent of ZF, or CH & GCH of ZFC, proceed by
constructing models where the other axioms hold but the axiom in question
fails. Since a conservative extension imposes additional requirements on the
models, it's not obvious that the independence proofs still go through.
For S4 + ZFC specifically, the models do not merely have to satisfy a couple
of additional modal propositional axioms. If S is the set of all Fs, we
define S's closure as everything that's possibly F, and its interior as
everything that's necessarily F. The comprehension axioms then require S's
closure and interior to exist, while the S4 axioms require closure and
interior, when defined in this way, to behave similarly to the topological
operations.
This by itself doesn't get us very far. I'm only suggesting there are open
questions here.
Mike Carroll
More information about the FOM
mailing list