Stationary subsets of P_kl with respect to the ground model
by
Natasha Dobrinen
Kurt Gödel Research Center for Mathematical Logic, University of Vienna

We are interested in the following problems: When is P_klÇV stationary and/or co-stationary in a model W extending V? and its cousin, If k ^{< k}=k in W, when is P_k⁺lÇV k-stationary and/or co-k-stationary? Preservation of the k-stationarity of P_k⁺lÇV is exactly what is needed to characterize certain distributive laws in Boolean algebras in terms of games. Although stronger, in a sense it is easier to see when k-stationarity of the gound model is preserved, due to a Kueker-type theorem for < k-ary functions. We present some conditions under which k-stationarity of the ground model is preserved. On the other hand, it seems to us easier to make P_k⁺l\V stationary than kappa-stationary, owing to the finite nature of first order logic, and even this requires large cardinals when k > w and no new w-sequences are added. Still, we are quite interested in making P_k⁺l\V k-stationary, as this is related to an open problem regarding games and distributive laws. Towards this, we present an equiconsistency result, due to Sy Friedman and ourselves, for when P_À2À₃\V is stationary.

Date received: March 28, 2005