TITLE: preservation of $\aleph_2$-c.c. in CS iterations
QUESTION [7 upvotes]: 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?

REPLY [7 votes]: One of the standard (counter)examples here has to do with uniformization at $\omega_2$, and it is dealt with in Section 3 of the Appendix to Shelah's Proper and Improper forcing:
Let $S=\{\delta<\omega_2:\rm{cf}(\delta)=\omega_1\}$, and for each $\delta\in S$ let $\eta_\delta:\omega_1\rightarrow\delta$ be the increasing enumeration of a closed unbounded subset of $\delta$, and let $\bar{\eta}=\langle \eta_\delta:\delta\in S\rangle.$
Let $\bar{C} =\langle c_\delta:\delta\in S\rangle$ be a collection of functions where $c_\delta$ maps $\omega_1$ to $\{0,1\}$.  We say that $\bar{C}$ can be uniformized if there is a function $f:\omega_2\rightarrow\{0,1\}$ such that for every $\delta\in S$ the set
$$\{\alpha<\omega_1: c_\delta(\alpha)\neq f(\eta_\delta(\alpha))\}$$
is bounded in $\omega_1$. 
Shelah proved that if CH holds, then for any such $\bar{\eta}$ there is a family of colorings $\bar{C}$ as above that cannot be uniformized.  Assaf Rinot has a nice write-up of this result on his blog.  
Shelah points out in Discussion 3.4A on page 984 of Proper and Improper forcing that for any given $\bar{C}$, there is a very nice $\aleph_1$-closed $\aleph_2$-cc (even $\aleph_2$-centered) forcing that adds a function uniformizing $\bar{C}$.
The relevance to your question is that an iteration of such uniformizing forcings that takes care of all potential $\bar{C}$ for a given $\bar{\eta}$ must collapse $\aleph_2$ as otherwise we contradict the theorem above, and therefore the limit of the iteration certainly cannot have the $\aleph_2$-cc.
The $\aleph_2$-pic (properness isomorphism condition) from Chapter VIII of Proper and Improper Forcing is a strong version of the $\aleph_2$-cc that says (roughly) that our forcing isn't doing something like "uniformizing", and it is one way to ensure that the limit of a countable support iteration possesess the $\aleph_2$-cc.