Forcing notions in inner models

There is a partial order ({mathbb{P}}) preserving stationary subsets of ω ₁ and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω ₁ over V also collapses ω ₁ over ({V^{mathbb{P}}}) . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V[c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B.