Is every set class generic over a given inner model?

Question

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 Vopěnka [Vopěnka and Hájek, 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)?

Answer 1

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.

Answer 2

What you are referring to it, is called the "extender algebra". It is a Boolean algebra and so can be considered as a forcing notion.

Then the result you have stated might be the following theorem of Woodin:

Theorem (Woodin). Assume $(M;\vec{E})$ is fully iterable and $\vec{E}$ witnesses a countable ordinal $\delta$ is a Woodin cardinal in $M$. Then for every set of ordinals $x$ there is a (well-founded) iteration $j: M \to M^*$ of length < $card(x)^+$ such that $x$ is $j(W_{\delta,\delta}(\vec{E}))-$generic over $M^*$.

For undefined notions and a proof of this theorem see the following papers:

Farah, "The extender algebra and $\Sigma^2_1 $-absoluteness".

Schindelr-Doebler, "The extender algebra and vagaries of $\Sigma^2_1 $-absoluteness''.

There is also another result of Woodin which is related to your question.

Theorem (Woodin) Work in $ZF$. Suppose the $HOD$ Conjecture is $\Omega-$valid from $ZFC$ + “There is a supercompact cardinal”, $\delta_0$ is a supercompact cardinal, and that there is a supercompact cardinal below $\delta_0$.

Then there exists a transitive class $N ⊂ V$ and $X ∈ V_{\delta_0}$ such that the following hold:

(1) $N \models ZFC$.

(2) $N$ is $Σ_2-$definable from $X$.

(3) There exists a partial order $P ∈ N ∩ V_{\delta_0}$ such that for all $A ⊂ Ord, A ∈ N[G]$ for some $N-$generic filter $G ⊂ P.$

Answer 3

1) I beleive that your formulation is incorrect. Solovay's Conjecture (SC) is: For every a (a set of ordinals) such that in L[a] zero-sharp does not exist then a is set generic. By Jensen's coding theorem SC is false.

2) For a proof of the above see page 4 in "Coding the Universe" by Beller, Jensen and Welch. It is also recommened to start with the two reviewes of this book by Friedman and Mitchell.