On a generalization of Jensen's □ κ , and strategic closure of partial orders

Introduction. It is well known that many statements provable from combinatorial principles true in the constructible universe L can also be shown to be consistent with ZFC by forcing. Recent work by Shelah and Stanley [4] and the author [5] has clarified the relationship between the axiom of constructibility and forcing by providing Martin's Axiom-type forcing axioms equivalent to 0 and the existence of morasses. In this paper we continue this line of research by providing a forcing axiom equivalent to ],. The forcing axiom generalizes easily to inaccessible, non-Mahlo cardinals, and provides the motivation for a corresponding generalization of El, In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = is a partial order. For each ordinal a we will consider a game GP played by two players, GOOD and BAD. 'The players choose, in order, the terms in a descending sequence of conditions . GOOD chooses all terms pA for limit A, and BAD chooses all the others. BAD wins if for some limit j has no lower bound. Otherwise, GOOD wins. Of course, we will be rooting for GOOD. A strategy for GOOD is a function Sa such that for any limit ordinal: , Sa(P) is a (possibly empty) set of lower bounds for n-in other words, a set of possible next moves for GOOD. If P = is a partially completed game of Ga, we say that GOOD has played by Sa if for each limit ordinal 3 in which GOOD has played by Saw Sa(p) # 0. Recall that El, is the statement that there is a sequence such that Ca is c.u.b. in a, if P is a limit point of Ca then CA = CaIn , and if cf(a) I a such that for all limit ordinals a < X: (1) Ca is c.u.b. in a.