  Shelah shows that certain revised countable support (RCS) iterations do not
add reals. His motivation is to establish the independence (relative to large
cardinals) of Avraham's problem on the existence of uncountable
non-constuctible sequences all of whose proper initial segments are
constructible, Friedman's problem on whether every 2-coloring of
$S^2_0=\{\alpha<\omega_2\colon\cf(\alpha)=\omega\}$ has an uncountable
sequentially closed homogeneous subset, and existence of a precipitous normal
filter on $\omega_2$ with $S^2_0\in{\cal F}$.
  The posets which Shelah uses in these constructions are Prikry forcing, Namba
forcing, and the forcing consisting of closed countable subsets of $S^*$ under
reverse end-extension, where $S^*$ is a fixed stationary co-stationary subset
of $S^2_0$. Shelah establishes different preservation theorems for each of
these three posets (the theorem for Namba forcing is particularly intricate).
We establish a general preservation theorem for a variant of RCS iterations
which includes all three posets in a straightforward way.
