Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without collapsing the continuum?

Question

Suppose the continuum is larger than $\aleph_2$. Does there exist a countably closed notion of forcing that collapses $\aleph_2$ to $\aleph_1$, but does not collapse the continuum to $\aleph_1$? Moreover, does there exist such a forcing notion that is separative and has size continuum? It is known (see below) that the canonical collapse Coll$(\aleph_1, \aleph_2)$ collapses the continuum. Trying something like the canonical collapse relativized to some inner model will fail to answer the question, because this forcing will not be countably closed in V.

Background information:

This question came up as a result of my studies of the following theorem.

Let $\kappa < \theta$ be cardinals, with $\kappa$ regular and $\theta^{<\kappa} = \theta$. Then any forcing of size $\theta^{<\kappa}$ which is separative and $<\kappa$ closed and which collapses $\theta$ to $\kappa$ is forcing equivalent to the canonical collapse forcing Coll$(\kappa, \theta)$.

I want to know whether this theorem still holds in the case where $\theta^{<\kappa} = \theta$ fails. The question above is the simplest possible such case.

The reason why Coll$(\aleph_1, \aleph_2)$ collapses the continuum (when CH fails) is that we can think of $\aleph_1$ as $\aleph_1$ many $\aleph_0$-blocks. Consider only the elements of Coll$(\aleph_1, \aleph_2)$ such that on each $\aleph_0$ block, they are either fully defined or fully undefined. This is a dense set in Coll$(\aleph_1, \aleph_2)$, and it's isomorphic to Coll$(\aleph_1, \aleph_2^{\aleph_0})$ = Coll$(\aleph_1, \bf{c})$.

Answer 1

Here is for the benefit of the curious (and it answers the original question): By a result of Baumgartner-Taylor (Saturation properties of ideals in generic extensions. I) we know $[\omega_2]^\omega$ can be splitted into $2^\omega$ many disjoint stationary subsets. For any proper forcing $\mathbb{P}$ that collapses $\omega_2$, we know in the generic extension there exists a club subset of $[\omega_2^V]^\omega$ of size $\omega_1$. Since stationary sets are preserved when forcing with proper $\mathbb{P}$, we can define an injection from $(2^\omega)^V$ into $\omega_1$ in $V[G]$.

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(For completeness) For the case of semi-proper forcings, the following will give the result:

1. We can partition $[\omega_2]^\omega$ into $2^\omega$ disjoint stationary subsets $S_r$ such that for each club $C\subset [\omega_2]^\omega$, $|S_r\cap C|\geq 2^\omega$ (just greedily union up the partition obtained from Baumgartner-Taylor)
2. (Todorcevic) For each stationary $S\subset [\omega_2]^\omega$, we can find its stationary core $S^*\subset S$ such that $|S-S^*|\leq \aleph_2$ and if $S^*$ is stationary then it remains so after any semiproper forcing. (the proof is pretty clever but I believe something written is coming out)
3. Assuming $2^\omega > \omega_2$, and apply 2 to each $S_r$ we see by the consideration of cardinality that the hypothesis is satisfied. But then we proceed as in the proper case.

Answer 2

I got the answer from Stevo Todorcevic last weekend at the MAMLS conference in honor of Richard Laver in Boulder, CO. He told me that it is an unpublished result of his that any semi-proper forcing which collapses $\aleph_2$ collapses the continuum. As countably closed forcing is semi-proper, the answer to my question is no. Stevo sketched a proof for me, but I do not remember it well enough to reproduce it here.

Answer 3

It might be late, but the following paper by Todorcevic analyzes the above question for three kind of forcing notions: Proper forcings, semi-proper forcing and forcings that preserve stationary subsets of $\omega_1$: Collapsing $ω_2$ with semi-proper forcing

Answer 4

Here is, I think, a partial answer. I believe I can show that as long as a countably closed forcing adds a new $\omega_1$-sequence, the continuum is collapsed below the size of the poset. I am not sure if you can do better.

Prop. Let $\mathbb{P}$ be a countably closed notion of forcing such that $\Vdash\dot{f}:\omega_1\rightarrow ON,\dot{f}\not\in V$. Then, if $G$ is $\mathbb{P}$-generic over $V$ we will have $V[G]\vDash 2^\omega\leq |\mathbb{P}|$.

Pf: It's enough to show that in $V[G]$, $|\mathcal{P}(\omega)\cap V|\leq |\mathbb{P}|$ (because $\mathbb{P}$ is countably closed). Note that for each $p\in\mathbb{P}$ there is some $\alpha<\omega_1$ such that $p$ doesn't decide $\dot{f}(\alpha)$ (otherwise $f$ could be defined in $V$); let $\alpha(p)$ denote the least such $\alpha$. Let $\beta_0(p)<\beta_1(p)$ be the least ordinals $\beta$ such that there's $q\leq p$ for which $q\Vdash\dot{f}(\alpha(p))=\beta$.

Fix in $V$ a well-ordering $\prec$ of $\mathbb{P}$. Now, working in $V[G]$, we associate to each $q\in\mathbb{P}$ an $x_q\subseteq\omega$ as follows. Inductively define a descending sequence of conditions $q_0\geq q_1\ldots \geq q_n\ldots $ by $q_0=q$, $q_{n+1}$ is the $\prec$-least member of $G$ below $q_n$ which decides $\dot{f}(\alpha(q_n))$. Let $x_q= \{n\in \omega|f(\alpha(q_n))=\beta_0(q_n)\}$ .

To finish we just have to show that for each $x\in\mathcal{P}(\omega)$ that the set $D_x=\{r\in\mathbb{P}|r\Vdash(\exists q\in \dot{G})x=\dot{x_q}\}$ is dense. Let $p\in\mathbb{P}$ be a fixed condition. Inductively define $p_0\geq p_1\geq \ldots p_n\geq $. Set $p_0=p$. If $n\in x$ set $p_{n+1}$ to be the $\prec$-least member of $\mathbb{P}$ with $p_{n+1}\leq p_n$ and $p_{n+1}\Vdash\dot{f}(\alpha(p_n))=\beta_0(p_n)$; if $n\not\in x$ then do the same thing but have $p_{n+1}\Vdash\dot{f}(\alpha(p_n))=\beta_1(p_n)$. Then let $r$ be below all the $p_n$. Then $r\in D_x$, with $p$ as our witnessing $q$.