[FOM] inverse forcing

Ali Enayat ali.enayat at gmail.com
Fri Jul 23 19:52:27 EDT 2010

In a recent posting (July 22), Monroe Eskew posed the following two questions:

> Let N be a transitive model of ZFC.  Let P be a partial order and G
> such that for some M, G is (M,P)-generic and N=M[G].
> 1) Is M unique?
> 2) Is M definable in N?

In a more recent posting (July 23), Eskew noted that the answer to (1)
in negative by choosing P as a collapsing poset.

Alternatively, one can choose mutually generic Cohen reals r and s
over some M, and observe that if N = M(r,s), then N is a generic
extension of each of M(r) and M(s), via the same notion of forcing.

However, the answer to (2) - assuming that one is allowed to use
parameters in the defining formula - is positive and nontrivial. This
result is independently due to Richard Laver and Hugh Woodin; for a
proof see Theorem 8 of the following paper:

Reitz, Jonas , "The ground axiom", Journal of Symbolic Logic 72 (4):
1299–1317 (2007).

Let me close by noting that it is easy to see that the above use of
parameters is necessary, e.g., in the example above, M(r) is not first
order definable by a parameter free formula in M(r,s).


Ali Enayat

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