Extending Namba forcing to arbitrary lengths

Question

Namba forcing is stationary-preserving and forces $cf(\omega_2^{\mathbf{V}}) = \omega$. Ronald Jensen used $\mathcal{L}$-forcing to iterate Namba posets in order to solve the extended Namba problem: for any strongly inaccessible $\kappa$, he constructed a stationary-preserving forcing notion $\mathbb{P}_{\kappa}$ that forces all ground model regular cardinals in the interval $(\omega_1, \kappa)$ to have cofinality $\omega$, while preserving the cofinality of $\kappa$. His construction allows $\mathbb{P}_{\kappa}$ to add no reals.

To formalise my question, let me use $Nb(\alpha, \beta)$ to denote the statement

$``$there is a stationary-preserving forcing notion $\mathbb{P}$ such that

• for all $\lambda \in \mathbf{V} \cap (\alpha, \beta)$, if $\lambda$ is regular, then $\Vdash_{\mathbb{P}} cf(\lambda) = \omega$, and
• $\Vdash_{\mathbb{P}} cf(\beta) > \omega$.$"$

In solving the extended Namba problem in the affirmative, Jensen showed that $Nb(\omega_1, \beta)$ is true when $\beta$ is strongly inaccessible. In another example of $\mathcal{L}$-forcing he gave, he used a modified construction to get $Nb(\omega_1, \beta^+)$ for any cardinal $\beta$ of cofinality $\omega_1$. The forcing notions he presented have the additional property of not adding reals, which I do not require.

My question is thus, is it known if $Nb(\omega_1, \beta)$ is true for all regular cardinals $\beta$?

Answer 1

An alternative to Andreas’ reply in the comments; using side condition methods one can construct forcings that perform similar jobs as the forcings by Jensen that were mentioned. The following is provable in ZFC by such rather elementary tools making use of countable models as side conditions:

Theorem. For every regular cardinal $\lambda$ and every set $\mathcal{K} \subset \lambda$ that consists of uncountable regular cardinals different from $\omega_1$, there exists a forcing $\mathbb{P}$ with the following properties

1. $\mathbb{P}$ is ssp,
2. for every $\kappa \in \mathcal{K}$, forcing with $\mathbb{P}$ changes the cofinality of $\kappa$ to $\omega$,
3. for every regular cardinal $\kappa \in (\omega_1, \lambda] \setminus \mathcal{K}$, forcing with $\mathbb{P}$ changes the cofinality of $\kappa$ to $\omega_1$.

Working with Boban Velickovic, I consider such a forcing in my thesis (forthcoming), in the context of constructing combinatorial analogues to $\mathcal{L}$-forcings. The conditions are basically finite configurations of suitable countable elementary substructures of some $H_\theta$, together with working parts that grow into countable sequences which are cofinal in the cardinals $\kappa \in \mathcal{K}$. To infer that $\mathbb{P}$ is ssp, one needs to know that there exist projective stationary many of these suitable models, this can be proved using arguments involving games similar to those in Section 3 of

Foreman, M., Magidor, M. Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on $P_\kappa(\lambda)$. Acta Math. 186, 271–300 (2001).

This particular forcing $\mathbb{P}$ does always add reals.