[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 

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