Questions about $\aleph_1-$closed forcing notions

Question

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:

1) $GCH$ fails everywhere,

2) there are no inaccessible cardinals,

3) there are no $\kappa-$Souslin trees,

4) Any non-trivial $c.c.c.$ forcing adds a real,

5) Any non-trivial $\kappa^+-$closed forcing notion collapses some cardinals.

Consistency of (1) is proved by Foreman-Woodin, (2) clearly can be consistent and the consistency of (4) is shown in "Forcing with c.c.c forcing notions, Cohen reals and Random reals".

My interest is in the consistency of (5). Let's consider the case $\kappa=\omega.$

Question 1. Is it consistent that any non-trivial $\aleph_1-$closed forcing notion collapses some cardinals?

The above question seems very difficult, and it is not difficult to show that for its consistency we need some very large cardinals. But maybe the following is simpler:

Question 2. Is it consistent that any non-trivial $\aleph_1-$closed forcing notion of size continuum collapses some cardinals? Does its consistency imply the existence of large cardinals?

Answer 1

The answer to the second question is yes, without any large cardinals assumptions.

Claim: if $2^{\aleph_0}$ is singular then every non-trivial $\sigma$-closed forcing of size $2^{\aleph_0}$ collapses the continuum.

The reason is that such forcing must add a new set of ordinals of size $\lambda < 2^{\aleph_0}$, $\tau$ (since the minimal $\lambda$ must be regular). I want to code $2^\omega$ into $\tau$.

If we choose $\lambda$ to be minimal, every initial segment of $\tau$ is in $V$. Now, we can define a function $F\in V$ from all possible initial segments of $\tau$ onto the reals, such that for every $p\in P$ and every $x\in 2^\omega$ there is $q\leq p$ and $\beta < \lambda$ such that $q\Vdash \tau \restriction \beta = \check{a}$ and $F(a)=x$:
Let $\langle (p_i, x_i) | i < 2^{\aleph_0}\rangle$ enumerate $P\times 2^\omega$. For every $\alpha < 2^{\aleph_0}$, we use the $\sigma$-closure of $P$ and the fact that $\tau \notin V$ in order to find $q \leq p_\alpha$ such that $q \neq p_i$ for every $i < \alpha$, $q\Vdash \tau \restriction \beta = \check{a}$ (for some $\beta$) and $F(a)$ is not determined yet, and set $F(a)=x_\alpha$:

Let $p = p_\alpha \in P$. We start by building a tree of $2^{<\omega}$ incompatible conditions $q_s,\, s\in 2^{<\omega}$ such that $q_\emptyset = p$ and for every $s\in 2^{<\omega}$, $q_{s\frown (0)}, q_{s\frown (1)} \leq q_s$, there is $\beta_s < \lambda$ in which $q_{s\frown (i)} \Vdash \tau \restriction \beta_s = a_{s\frown (i)}$ for $i\in\{0,1\}$, $a_{s\frown (0)} \neq a_{s \frown (1)}$. This is possible since $\tau \notin V$ but every initial segment of it is in $V$.

For every $f\in 2^\omega$, let $q_f \in P,\,\forall n\, q_f \leq q_{f\restriction n}$ (by the closure of $P$). Without loss of generality, $\forall f \in 2^\omega\,q_f\Vdash \tau \restriction \beta = \check{a_f}$, and for every $f\neq f^\prime$, $a_f \neq a_{f^\prime}$ (take $\beta = \sup_{s\in 2^{<\omega}} \beta_s$). Since we picked already only $|\alpha |<2^{\aleph_0}$ values, there must be some $f\in 2^\omega$ such that $F(a_f )$ is not determined.

By density arguments, in $V[G]$, $\{F(\tau \restriction \beta) | \beta < \lambda\} = (2^\omega)^V$.

Answer 2

I can't answer your question, but I can give a simpler sounding formulation that might be helpful. Analyze the question in two cases.

Case 1: The continuum hypothesis holds
In this case, the statement is false, because any $<\aleph_1$ -closed forcing of size $\aleph_1$ cannot collapse cardinals. The forcing to add a cohen subset to $\aleph_1$ is a nontrivial example of such a forcing.

Case 2: The continuum hypothesis fails
In this case, it is a theorem that every $<\aleph_1$-closed forcing notion which collapses a cardinal collapses the continuum. (See this question, referenced by Joel in the comments.) But every such forcing is equivalent to the canonical collapse forcing to collapse the continuum to $\aleph_1$. The most general version of this latter theorem that I know of (although the degree of generality makes it hard to follow) can be found in Handbook of Boolean Algebras, Volume 2, Corollary 1.15.

So really, your question boils down to whether every $<\aleph_1$-closed forcing of size continuum is isomorphic to Coll$(\aleph_1, c)$ This sounds strange to me, but I can't prove it's false, and if Foreman is entertaining it, who am I to judge it?

Answer 3

The following results are obtained in a joint work with Yair Hayut and are now presented in our joint paper On Foreman's maximality principle:

Theorem 1. (Assuming the existence of a strong cardinal) There is a model of $ZFC$ in which for all uncountable (regular) cardinals $\kappa,$ any $\kappa$-closed forcing notion of size $\leq 2^{<\kappa}$ collapses some cardinals.

Note that in such a model $GCH$ should fail everywhere and hence some very large cardinals are needed. The following theorem is in the opposite direction:

Theorem 2 (Assuming the existence of a strong cardinal and infinitely many inaccessibles above it) There is a model of $ZFC$ in which $GCH$ fails everywhere and for each uncountable (regular) cardinal $\kappa,$ there exists a $\kappa$-closed forcing notion of size $2^{<\kappa}$ which preserves all cardinals.