[FOM] Showing that "forcing is the unique method"

Timothy Y. Chow tchow at alum.mit.edu
Mon Sep 24 16:54:57 EDT 2007

As I've mentioned before on FOM, Problem 2.4 in Shelah's article "Logical 
Dreams" is to show that forcing is the unique method (for independence 
proofs) in a non-trivial sense.

So are there known results of the following general form?  "If N is an 
extension of a model M (of ZF, say) satisfying conditions A, B, and C, 
then N must be a forcing extension of some kind."

In Cohen's book "Set Theory and the Continuum Hypothesis," he prefaces his 
treatment of forcing with some interesting negative results to show what 
kinds of things *won't* work.  I've found his discussion very illuminating 
and was wondering if there are other similar results, that show how one is 
almost (ahem) forced to use forcing in independence proofs.  Such results 
would also indicate what direction to look in if confronted with a 
statement that one feels might be independent of ZF but that seems 
unlikely to yield to a forcing argument.


More information about the FOM mailing list