[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

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