Who needs RCS iterations?

Question

According to this paper of Chaz Schlindwein, any countable support iteration of semi-proper forcings is semi-proper. This seems like a breakthrough simplification, and I wonder why it is not more well-known. Before diving into the arguments I want to ask the forcing community whether this work is widely viewed as correct? Thanks for your help.

Answer 1

I think that this claim is false, at least in the way that I understand "countable support". The following was explained to me by Menachem Magidor in the 1990s; it may be folklore, and I suspect it is the reason for introducing RCS in the first place.)

Consider the following countable support iteration $(P_\alpha, Q_\alpha: \alpha < \kappa)$ of length $\kappa$, where $\kappa$ is either a measurable cardinal or $\omega_2$ in a universe where Namba forcing is semiproper:

• $Q_0$ is Prikry forcing on $\kappa$, or Namba forcing.
• For each $\alpha>0$, $Q_\alpha$ is an antichain on $\omega_1^V$ (with an additional largest element $1_\alpha$ thrown in). (Really: the canonical $P_\alpha$-name for this set.)
• Each $Q_\alpha$ is semiproper. (Even proper, except for $Q_0$.)
• For $0\le \alpha\le \kappa$, $P_\alpha$ is defined in the usual way, as the set of all countable partial functions $p$ with domain $\subseteq \alpha$ such that for all $\beta\in dom(p)$ we have $p\restriction \beta$ is in $P_\beta$ and forces $p(\beta)\in Q_\beta$.

Then the following statements are true:

• Every condition in $P_\kappa$ has bounded domain. (This would not be true if we had used revised countable support iteration.)
• $Q_0$ introduces a new sequence $(\kappa_n: n \in \omega)$ which is cofinal in $\kappa$. (I.e., there is a $Q_0$-name for such a sequence. Hence there is also a $P_\kappa$-name for such a sequence.)

We define functions $g$ and $h$ as follows:

• Let $g$ be a $P_\kappa$-name for the function from $\kappa\setminus \{0\}$ to $\omega_1^V$ which assigns to each ordinal $\alpha$ the generic element of the antichain $Q_\alpha$.
• Let $h$ be the $P_\kappa$-name for the function $n\mapsto g(\kappa_n)$.

Then the empty condition forces that $h$ is a surjective function from $\omega$ onto $\omega_1$. (Hence $P_\kappa$ is not semiproper.)

Proof: Assume that $\gamma< \kappa$, and that $p\in P_\kappa$ is a condition. It is enough to find a stronger condition forcing that $\gamma$ is in the range of $h$.

First find $\alpha$ such that $p\in P_\alpha$.
Note that $p$ "knows nothing" about the values of $g$ above $\alpha$, and that $p$ knows very little about the sequence $(\kappa_n:n\in\omega)$.

Let $n$ be the length of the stem of $p(0)$, the $Q_0$-component of $p$. By extending the stem, we can find $p'(0)$ stronger than $p(0)$, such that $p'(0)$ forces a value (say $\beta$) to $\alpha_n$, and such that $\beta> \alpha$.

Define $p'$ as follows: $dom(p')=dom(p)\cup \{\beta\}$. $p'(\beta)=\gamma$. (And $p'(i)=p(i)$ for all $i\not=0,\beta$.)

Now $p'$ forces that $h(n)=g(\alpha_n)=g(\beta)=\gamma$. QED

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In defense of Schlindwein's paper I should add that he might have had a different (nonequivalent) definition of "CS iteration" in mind.

Answer 2

Following up on Martin's answer:

Definition 10 of the paper defines the concept "hemi-properness", shows that semi-proper forcings are hemi-proper, and then claims (Theorem 16) that hemi-properness is preserved under countable support iterations. This last is demonstrably false, as we will show that "hemi-properness" is equivalent to "preserves $\omega_1$", and it is well known that this property is not preserved by countable support iteration.

Definition [Definition 10 from the paper]

A poset $P$ is hemi-proper if and only whenever $\lambda$ is a sufficiently large regular cardinal, $M$ and $N$ are countably elementary submodels of $H(\lambda)$, $P\in M\in N$, and $q\in P\cap M$, then $q\nVdash``\omega_1\cap M[G_P]>\omega_1\cap N$''.

Lemma 11 in his paper shows that hemi-proper forcings do not collapse $\omega_1$.

Claim: If forcing with $P$ preserves $\omega_1$, then $P$ is hemi-proper.

Let $\lambda$, $M$, $N$, and $q$ be given. Any condition forces that $M[G_P]$ is countable (it is just the interpretations of the countably many $P$-names living in $M$). In particular, any condition forces that $M[G_P]\cap\omega_1$ is an ordinal $\alpha<\omega_1$ since $\omega_1$ is preserved. The model $N$ can see $M$, and so $N$ will contain a name $\dot\alpha$ for the countable ordinal $M[G_P]\cap\omega_1$ from $M[G_P]$. In $N$, we can extend $q$ to a condition $r$ deciding a particular value $\alpha$ for $\dot\alpha$. Notice that since $r$ and $\dot\alpha$ are in $N$, the countable ordinal $\alpha$ is in $N$ as well. Necessarily we have $\alpha<N\cap\omega_1$. Since $r$ extends $q$, it follows that $q$ did not force $\omega_1\cap M[G_P]$ to be greater than $N\cap \omega_1$. This establishes that $P$ is hemi-proper.

Schlindwein lectured on this at a few meetings in the 1990s. The paragraph at the top of page 9 was added in an attempt to meet my objections that his notion was the same as "preserves $\omega_1$'', but there are no such $M$ and $N$ as he describes there.

Chaz was a nice guy. I'm sad to hear the news that he has passed away.