There are a few classical theorems in set theory:
- The finite support iteration of ccc forcing is ccc.
- The countable support iteration of proper forcing is proper.
- The finite support iteration of length $\omega$, Cohen forcing is a Cohen forcing.
But what happens when you iterate Cohen forcing with a countable support iteration of length $\omega$?
What do we already know:
- This is a proper forcing. Because Cohen forcing is proper.
- This is not a ccc forcing, because we iterate with full support.
What can you say about the full/countable support iteration of length $\omega$ of Cohen forcings?
Specifically, does it add non-Cohen reals? Does it collapse any cardinals?