TITLE: Who needs RCS iterations?
QUESTION [15 upvotes]: 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.

REPLY [13 votes]: 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.