[FOM] What can't be forced?
joeshipman at aol.com
Mon May 21 00:31:20 EDT 2007
Forcing may be regarded as a technique for obtaining relative
consistency results with ZFC, by starting with a countable transitive
model M of ZFC and extending it by a "generic set" G in such a way that
M(G) |= ZFC and M(G) |= Phi. More generally, one can start with a
countable transitive model of ZFC + Psi, where Psi is a large cardinal
axiom, in order to build an M(G) that satisfies both ZFC and Phi.
Are there any statements known to be consistent with ZFC [assuming the
existence of a CTM] which cannot be obtained in this way?
Of course, taking the question literally, any statement which may hold
in a CTM can be "forced" by *starting* with that CTM, so the question
reduces to "are there any statements consistent with ZFC which cannot
hold in any CTM's"?
A more precise question to ask is "if we start with the minimal
countable transitive model M, which consists of L(alpha) for the first
alpha where this gives a model of ZFC, is there any statement known to
be consistent which does not hold in M(G) for any generic set G"?
AOL now offers free email to everyone. Find out more about what's free
from AOL at AOL.com.
More information about the FOM