[FOM] inverse forcing problem

[Monroe Eskew]

The answer is no.  There are some simple counterexamples:

Let \kappa < \lambda be uncountable regular cardinals, with \kappa >
2^{\omega} in V.  Let P = Coll(\omega,\lambda), Q =
Add(\omega,\kappa), R = Add(\omega,\omega).  Let G be P-generic over
V.  In V[G] there is a bijection f between \kappa and \omega.   let H
be Q-generic over V[G], and let H* be the image of the isomorphism
between Q and R induced by f.  Then V[G][H] = V[G][H*] = V[H][G] =
V[H*][G].  Clearly V[H] \not= V[H*].

Now is there any kind of forcing from which we can recover V as a
subclass of V[G], given G?

On Thu, Jul 22, 2010 at 11:42 AM, Monroe Eskew <meskew at math.uci.edu> wrote:
> 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?
>