# Is every set class generic over a given inner model?

In a paper by B. Mitchell, I stumbled into the following sentence:

"In the summer of 1986 Woodin discovered the second of the forcing orders associated with a Woodin cardinal, the extender algebra. This forcing goes back to the class forcing of $Vop\check{e}nka$ [$Vop\check{e}nka$ and $H\acute{a}jek$, 1972], by which any set is generic, by a class forcing, over any given class model of set theory."

If I interpreted the result correctly, it means that for every inner model $M$ and for every set $x\in V$, there is a class forcing notion $\mathbb{P}$, definable over $M$, s.t. $x$ is $\mathbb{P}$-generic over $M$.

I looked up the reference, which was the book "The theory of semisets", but it was really hard to figure out anything because of the uncommon symbolization.

My questions are the following:

1) Is my formulation of this result correct?

2) Is there another, more approachable, reference where I could find its proof (and maybe more information on class forcing)?

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The questions have been answered already, but you may want to check as well the work of Sy Friedman, and his text on "Class Forcing". He proves that, appropriately formulated, the answer is no (in the presence of suitable large cardinals) even if we allow "hyperclass" forcing. –  Andres Caicedo Jul 10 '12 at 20:26
+1 Andres. The title of Sy Friedman's book is Fine Structure and Class Forcing (Walter de Gruyter, 2000) - books.google.com/books/about/… –  François G. Dorais Jul 10 '12 at 21:12
Thanks Andres! This gives a negative answer to my question (at least when $0^{\sharp}$ exists). I guess Mitchell is referring to the theorem mentioned by Peter. –  Lstoa Jul 11 '12 at 13:48

Isn't it Theorem 15.46 in Jech's Set Theory (Springer 2003) book? Perhaps one can reformulate it as follows: every set is in some generic extension of HOD.

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But how can one derive from this theorem, that this is true for every inner model $M$? –  Lstoa Jul 10 '12 at 19:57
Sorry, I don't know. –  Péter Komjáth Jul 10 '12 at 20:01
In any case thanks for your answer! At least it gives an example of such an inner model (it even works with set forcing, something which is not true for $L$ as Eran mentioned!). –  Lstoa Jul 10 '12 at 20:08