Preservation Results for Iterations of Non-Proper Forcing

Question

Suppose $\mathbb{P}$ is a forcing with the following properties: Let $G \subseteq \mathbb{P}$ be filter generic over $V$, then there exists $A \in V[G]$ such that $V[G]$ thinks $A$ is countable and $A \subseteq {}^\omega 2 \cap V$, but $A$ is not covered by any ground model countable set. That is, in the generic extension there is a countable set of ground model reals that can not be covered by a countable set from the ground model.

Certainly $\mathbb{P}$ is not a proper forcing.

Is it possible that $\mathbb{P}$ can preserve $\aleph_1$? As $A$ need not be a set in $V$, it seems at least theoretically possible that $\mathbb{P}$ does not need to collapse any cardinals.

Now I would like to iterate this forcing by itself. However, since it is not proper, I do not know what properties can be expected to be preserved.

Although this may be a bit too open ended, my next question is : What classes of forcing (by which I means things like c.c.c., proper, semi-proper, etc) could $\mathbb{P}$ potentially belong to given that it adds a countable set of ground model real that can not be covered by a countable ground model set. (Certainly $\mathbb{P}$ can not be c.c.c. or proper.)

Do any of these classes have preservation theorems for iterations (countable support or perhaps some other type of iterations)? I am most interested in perserving $\aleph_1$, preserving ${}^\omega\omega$-bounding, or $\aleph_2$-chain conditions. If it is applicable, one may assume $\mathbb{P}$ has size $\aleph_1$ if $\mathsf{CH}$ holds.

I am looking for some class of forcing that can help handle iterations of non-proper forcings like $\mathbb{P}$. Thanks for any information that can be provided.

Answer 1

The existence of a forcing notion $\mathbb{P}$ like that is equivalent to $\neg\text{CH}$.

On the one hand, Noah's comment shows that if CH holds, then one cannot add a countable set of ground-model reals that is not covered by any countable ground model set, while preserving $\omega_1$.

Conversely, suppose that CH fails in $V$, and consider Namba forcing. This adds a new cofinal $\omega$-sequence in $\omega_2^V$, and hence because $\neg\text{CH}$ it adds a countable set of ground-model reals, which is not covered by any ground-model countable set, or indeed by any ground model set of size $\omega_1$ in the ground model, but it preserves $\omega_1$. QED

It seems to be consistent relative to large cardinals that Namba forcing is semi-proper, in which case one might hope to iterate it.

Here is another kind of example. Start in $V$ with a measurable cardinal $\kappa$, and let $V[G]$ be the extension obtained by adding $\kappa$ many Cohen reals, so that $2^\omega=\kappa$ in $V[G]$. Now, in $V[G]$, let $\mathbb{P}$ be Prikry forcing for a normal measure $\mu$ in $V$, as defined in $V$. If $H\subset\mathbb{P}$ is $V[G]$-generic, then we can view $V[G][H]$ as $V[H][G]$, that is, as first doing Prikry forcing, and then adding reals. So all cardinals are preserved. But the Priky sequence will add a new $\omega$-sequence cofinal in $\kappa$, and hence a new countable set of reals from $V[G]$, which is not covered by any set in $V[G]$ of size less than $\kappa$ in $V[G]$. Thus, forcing over $V[G]$ with the Prikry forcing of $V$ has your desired properties.

And there are various complicated ways to iterate Priky forcing, depending on what you want to accomplish, and what kind of large cardinals you have available.

Answer 2

Chapters X, XI and XV (possibly also others) in Shelah's book "Proper and Improper Forcing" deal with the problem of iterating nonproper forcing notions. (By this I mean very non-proper. E.g., not even $S$-proper for any stationary $S$.)

• Chapter X gives a definition of revised countable support iteration (RCS). (It is somewhat difficult to read, but there are other definitions in the literature which could be used instead.
• Chapter XI deals with nonproper iterations not adding reals over models of CH. There is a property of forcing notions $Q$ that I will call $Pr_{X}(Q)$ that satisfies: Any forcing which has property $Pr_X(Q)$ preserves $\omega_1$ and moreover does not add reals. AND: The limit of an RCS iteration (that uses lots of collapses) of forcing notions with $Pr_X$ will itself have property $Pr_X$.
• Chapter XV introduces a property that I will call $Pr_{XV}$ and proves a similar statement: Any forcing which has property $Pr_X(Q)$ preserves $\omega_1$ AND: The limit of an RCS iteration (that uses lots of collapses) of forcing notions with $Pr_{XV}$ will itself have property $Pr_{XV}$.

A main point is that Namba forcing $Nm$ satisfies both of these properties. (Regardless of whether $Nm$ is semiproper or not. There are several versions of $Nm$: Laver-like or Miller-like, using the club filter or the cobounded filter; I am not sure if all of them have these properties.)

Roughly speaking, $Pr_{XV}(Q)$ is this: whenever you have a sufficiently nice tree $(N_\eta: \eta\in \omega_2^{<\omega})$ of countable elementary submodels of the universe, where niceness in particular implies that the intersections of the models with $\omega_1$ converge to the same ordinal $\delta$ along every branch, then $Q$ forces that there exists a branch $\nu\in \omega_2^\omega$ such that $N_\nu[G]\cap \omega_1=N_\nu\cap \omega_1=\delta$. As I recall, if $Q=Nm$ then the generic branch $\nu$ (the union of all stems of conditions in the generic filter) will satisfy the requirement.