[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.
Tim
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