[FOM] What can't be forced?
joeshipman@aol.com
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"?
-- JS
________________________________________________________________________
AOL now offers free email to everyone. Find out more about what's free
from AOL at AOL.com.
More information about the FOM
mailing list