2 Added implication between the two questions.

# A sequence of generic filterfilters that does not come from an iteration

Fix a countable transitive model $M$ of ZFC. In my answer to this question I indicated that there are forcing iterations $((Q_\alpha:\alpha\leq\omega),(\dot P_\alpha:\alpha<\omega))$ in $M$ and sequences $(G_\alpha:\alpha<\omega)$ of filters such that the following happens:

Each $G_\alpha$ is a filter in the evaluation of $\dot P_\alpha$ with respect to the filter $G_0*\dots*G_{\alpha-1}$ and $G_\alpha$ is generic over $M[G_0,\dots,G_{\alpha-1}]$ (call such a sequence $(G_\alpha:\alpha<\omega)$ a sequence of generics), but there is no $Q_\omega$-generic filter over $M$ whose $\alpha$-th projection is $G_\alpha$ for all $\alpha<\omega$.

An example can be obtained as follows:
Take the countable support iteration of Sacks forcing (or any other nontrivial $\omega^\omega$-bounding proper forcing notion) of length $\omega$ (i.e., the supports are actually everything, but this doesn't matter). This forcing adds no Cohen real.

Compare this to the finite support iteration of the same forcing notions. This iteration does add a Cohen real. The Cohen real is coded by the sequence of generics and hence this sequence of generics does not come from a generic filter for the countable support iteration mentioned before. This sequence of generics is not even contained in a forcing extension obtained using the countable support iteration.

Now here are two questions:

1) Is there an example of a sequence of generics (of length $\omega$) that cannot come from any iteration of the $\dot P_\alpha$?

I am asking here for iterations where the finite initial segments are as usual (just plain iteration) and we choose whatever ideal for the supports, including all finite subsets of the index set. But I am open to more general forms of iteration. For example take a large forcing notion $Q$ along with commuting complete embeddings of the $Q_\alpha$, $\alpha<\omega$. This would be an iteration of the $\dot P_\alpha$, too, the most general one that I can think out of right now.

2) Is there an example of a sequence of generics over $M$ that is not contained in any countable transitive extension of $M$ with the same ordinals as $M$ that is a model of ZFC?

Obviously, a positive answer to 2) solves 1) as well.

1

# A sequence of generic filter that does not come from an iteration

Fix a countable transitive model $M$ of ZFC. In my answer to this question I indicated that there are forcing iterations $((Q_\alpha:\alpha\leq\omega),(\dot P_\alpha:\alpha<\omega))$ in $M$ and sequences $(G_\alpha:\alpha<\omega)$ of filters such that the following happens:

Each $G_\alpha$ is a filter in the evaluation of $\dot P_\alpha$ with respect to the filter $G_0*\dots*G_{\alpha-1}$ and $G_\alpha$ is generic over $M[G_0,\dots,G_{\alpha-1}]$ (call such a sequence $(G_\alpha:\alpha<\omega)$ a sequence of generics), but there is no $Q_\omega$-generic filter over $M$ whose $\alpha$-th projection is $G_\alpha$ for all $\alpha<\omega$.

An example can be obtained as follows:
Take the countable support iteration of Sacks forcing (or any other nontrivial $\omega^\omega$-bounding proper forcing notion) of length $\omega$ (i.e., the supports are actually everything, but this doesn't matter). This forcing adds no Cohen real.

Compare this to the finite support iteration of the same forcing notions. This iteration does add a Cohen real. The Cohen real is coded by the sequence of generics and hence this sequence of generics does not come from a generic filter for the countable support iteration mentioned before. This sequence of generics is not even contained in a forcing extension obtained using the countable support iteration.

Now here are two questions:

1) Is there an example of a sequence of generics (of length $\omega$) that cannot come from any iteration of the $\dot P_\alpha$?

I am asking here for iterations where the finite initial segments are as usual (just plain iteration) and we choose whatever ideal for the supports, including all finite subsets of the index set. But I am open to more general forms of iteration. For example take a large forcing notion $Q$ along with commuting complete embeddings of the $Q_\alpha$, $\alpha<\omega$. This would be an iteration of the $\dot P_\alpha$, too, the most general one that I can think out right now.

2) Is there an example of a sequence of generics over $M$ that is not contained in any countable transitive extension of $M$ with the same ordinals as $M$ that is a model of ZFC?