According to [this paper](http://arxiv.org/pdf/math/9612208v1.pdf) 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.



Who needs RCS iterations?
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




---


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


