We consider algebras \(A\) on \(\aleph_\omega\), and look at possible mutual stationarity properties they may enjoy. ("Mutual stationarity" is a concept introduced by Foreman and Magidor in their discussion of the, generally, non-saturatedness of the non-stationary ideal.) We look to see whether a sequence of sets \(S_n\) each stationary below \(\aleph_{n+1}\) can be "simultaneously stationary" as a sequence for all algebras \(A\).
A second (but simpler for us) problem concerns a property and question of Pereira related to free subsets, internally approachable models, and the PCF conjecture.
Covering Lemma arguments using inner models provide lower bounds for both properties.