*Question:* Under great cardinal axioms, what is the intersection of all internal models? $ M $ of ZFC such that each set in $ V $ is set-generic about $ M $?

Each set belongs to a generic HOD extension, and we expect HOD to be canonical in the true $ V $, but ordinary large cardinal axioms do not imply that HOD is a canonical model of good behavior. Each accounting model $ M $ ZFC is HOD from some ZFC model (obtained using class forcing on $ M $)

However, let's $ M_∞ $ (or $ M_ text {Ord} $) be the minimum iterable internal model with a suitable class of Woodin cardinals. There is a definable ordinal iteration $ M & # 39; _∞ $ from $ M_∞ $ such that each set in $ V $ is set-generic about $ M & # 39; _∞ $. Specifically, choose an OD set of ordinals $ X_0 $; iterate the first Cardinal Woodin of $ M_∞ $ to make $ X_0 $ generic; then choose $ X_1 $ and iterate the second Cardinal Woodin to make $ X_1 $ generic, and so on. Also, by using generic character over local HOD, we can choose $ M & # 39; _∞ $ such that $ M & # 39; _∞∩H (λ) $ is definable in $ H (λ) $ (for $ λ> c $) and with each $ X⊂λ $ being $ M & # 39; _∞ $generic for a poset in $ M & # 39; _∞∩H ((2 ^ λ) ^ +) $ (as usual, $ H (λ) = {x: | mathrm {tc} (x) | <λ } $)

But is this optimal? For each $ M $ in the question and a set of ordinals $ s∈M $, it is $ M_∞ (s) $ elementally integrable in a $ M $-definable submodel of $ M $? Does the intersection of all those $ M $ same $ M_∞ $ with the least measurable cardinal iterated away? And what kind of great cardinals should such $ M $ to have?

Using $ ω $ induction steps of the central model, each $ M $ as in the question satisfies the projective determination (PD) in all the generic extensions of $ M $ (assuming PD in all generic extensions of $ V $), but I don't know how far central model induction can go here.

Please note that each set $ S $ is generic about some (depends on $ S $) iterate from $ M_1 $ (the minimal iterable internal model with a Cardinal Woodin). So for example if there is a super strong cardinal (and each set has a sharp edge) then there is a generic extension of $ M_1 $ with a super strong cardinal. This is analogous to the existence of complicated transitive models in $ L $; and more Woodin cardinals give models with more closure.

*Formalization note:* The answer is presumably the same regardless of whether $ M $ it is $ Σ_2 $ definable using parameters in $ V $, or we use NBG (plus large cardinal axioms) and try $ M $ as a class. Furthermore, allowing the choice to fail $ M $ The answer will likely not change. A large and likely cardinal assumption is that $ M_∞ $ (above) exists and is completely iterable.

*Local versions:* A variation is to consider internal models $ M $ with (for a specific $ λ $) each element of $ H (λ) $ generic over $ M $ using a force on $ H (λ) $. Examples include:

– for accounting purposes strong limit $ λ $, some iterations of $ M_ω $

– for singular strong limit $ λ $ of uncountable cofinality, some iterations of the minimal iterable internal model with a measurable number of Woodin cardinals (i.e. $ κ $ Woodin cardinals with $ κ $ measurable in the model)

– for inaccessible $ λ $, some iterations of $ M_∞ $.

*Class forcing:*

While part of the class forcing is similar to a set, the class forcing generally lacks the same type of closure. For example, even keeping ZFC, we can encode the universe into a real, even if each set is sharp. Lack of closure makes it easier to do $ V $ Generic, and if I understand it correctly, it is enough to use an appropriate iteration of the minimal internal model that satisfies "Ord is Woodin" (I'm not sure if we need its acute, or if there are definable issues), with the class forcing a string of Ord satisfactory condition (and therefore behaved well). An analogous relationship should also be valid for various extensions of the set theory language, with "Ord is Woodin" (and the internal model) and the closing properties of classes (and class forcing, or class forcing). Ord-cc) strengthened in the same way

While many iterations should work, a particularly fancy choice and encoding of an iteration is (conjecturally) the & # 39; stability & # 39; $ S = {n, α, β: n <ω ∧ H (α) ≺_ {Σ_n} H (β) } $. $ (L (S), ∈, S) $ it's called a stable core (see The stable core and the structural properties of the stable core). Warning (conjectural): the iteration encoded by $ S $ (There are different encodings, but if it works, between iterations in which $ S $ is definable, the only iteration that can be defined for each iteration in which $ S $ is definable) is out of $ (L (S), ∈, S) $although another iteration is $ text {HOD} ^ {L (S)} = K ^ {L (S)} $. Without big cardinal assumptions, the stable nucleus theory is not canonical, but we still get the generic character. While the specific choice of $ S $ It's somewhat arbitrary, I think the use of the cumulative hierarchy is important to genericity, and presumably a different definition would simply lead to a different iteration that works, or would be insufficient or suboptimal.