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

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0022481200037415How to cite this article: Dan Velleman (1983). 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 • * .In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = <P, <, > is a partial order. For each ordinal a we will consider a game G" played by two players, GOOD and BAD. 'The players choose, in order, the terms in a descending sequence of conditions (Pp\ ]3 < «>• GOOD chooses all terms/^ for limit /3, and BAD chooses all the others. BAD wins if for some limit fi<a, GOOD is unable to move at stage /3 because (p T \y < j3> has no lower bound. Otherwise, GOOD wins. Of course, we will be rooting for GOOD.A strategy for GOOD is a function S a such that for any limit ordinal /3 < a and any descending sequence p = (p r \f < j3>, S a (p) is a (possibly empty) set of lower bounds for p-in other words, a set of possible next moves for GOOD. If p = (p r \ y < |3> is a partially completed game of G£, we say that GOOD has played by S a if for each limit ordinal d < /3, Ps e S a (p [ d). S a is a winning strategy for GOOD if for each limit ordinal /3 < a and partially completed game p = <P T \r<P> i n w h i c h G O O D h a s played by S a , S a (p) # 0.Recall that n , is the statement that there is a sequence < C a | a < K + , a a limit ordinal) such that C a is c.u.b. in a, if /3 is a limit point of C a then C^ = C a |~| j3, and if cf(a) < K then type(C 0 ) < K. Here "c.u.b." abbreviates "closed and unbounded", and "type(C")" is the order type of C a . Our generalization of • is defined as follows: DEFINITION 1. For any infinite cardinal K, B(K) is the statement: There is a sequence « C a , O | a < K, a a limit ordinal) such that for all limit ordinals a < K:(1) C" is c.u.b. in a.

Approachability and Games on Posets

“…As for G n -type games, Velleman [11] has shown that for any infinite cardinal K, every cr-closed, (K + 1)-strategically closed poset is K + -strategically closed if and only if D K holds. Ishiu and Yoshinobu [5] have shown that the condition 'a-closed' can be dropped in Velleman's result.…”

“…Note that D w is provable in ZFC. D K can be considered as a property of K + , rather than of K. From this point of view Velleman [11] generalized the notion of square principles, enabling us to mention its 'limit cardinal analogue'. DEFINITION 2.2.…”

“…It is easy to see that for any infinite cardinal K, D K is equivalent to B(K + ). The following are basic facts on B{k): [11]). Let X be any regular uncountable cardinal.…”

Jensen's ⃞ principles and the Novák number of partially ordered sets

“…However we know that this may fail in the above models. Namely, using a result of Laver it was observed by Velleman in [16] that it is consistent that there is a supercompact cardinal K and a set S £ K + such that {a < K + |cof a = K} £ S and D s holds. Now both the Levy collapse of K and the standard poset forcing PFA are K-CC, SO the class of ordinals of cofinality K remains the same in the extension and our sequence is still a Ds sequence.…”

“…The following is a slight modification of the partial order for adding a D W2 sequence with countable conditions, see e.g. Velleman [16]. EXAMPLE Now it is straightforward to check that 3P is tr-closed and, if CH holds, also satisfies the K 2 -cc.…”