It seems that considerable care is taken in the literature to ensure that a countable support (CS) iteration of proper forcings preserves $\aleph\_2$. Can you give an example, assuming CH, of a CS iteration of countably closed forcings such that each iterand is forced to have the $\aleph\_2$-c.c. but the full iteration does not? 



preservation of $\aleph_2$-c.c. in CS iterations
Assume CH. We start with a model with a function $F:[\omega\_2]^2\to\omega$ 
such that the following holds: if $\{A\_\alpha:\alpha<\omega\_2\}$ 
are pairwise disjoint countable subsets of $\omega\_2$ and $i<\omega$ then 
there are $\alpha<\beta$ such that all values of $F$ between $A\_\alpha$ 
and $A\_\beta$ are $>i$. This can be forced wih countable approximations. 


Given $P\_n$ let $q\in Q\_n$ iff $q$ is a countable subset of $\omega\_2$ 
with all values of $F$ on $q$ differing from $n$. $P$ is the full support 
iteration of the $Q\_n$'s. 


First it is easy to see that the conditions $p\in P\_n$ are dense where 
$p(i)$ is a real countable set for $i<\omega$. Using this, we can show that 
$Q\_n$ is $\aleph\_2$-cc, for all $n$. With the usual trick, this gives a 
$\gamma\_n<\omega\_2$ such that if we remove the part of $Q\_n$ below $\gamma\_n$, 
then we get a notion of forcing (=no minimal elements). 


Clearly, all $Q\_n$'s are countably closed. 


However, if $p\_\alpha$ is the condition with $p\_\alpha(n)=\{\alpha\}$, 
then any two of the $p\_\alpha$'s are incompatible. 


