Simplified morasses with linear limits

Abstract

In a recent series of papers Kanamori ([4], [5], and [6]) defines generalizations of several combinatorial principles known to follow from the existence of morasses. Kanamori proves the consistency of his generalizations by forcing arguments which come close to satisfying the hypotheses of the Martin's Axiom-type characterizations of morasses developed independently by Shelah and Stanley [9] and the author [12]. A similar “almost application” of morasses appears in [11], in which Todorčević uses forcing to prove the consistency of the existence of Kurepa trees with no Aronszajn or Cantor subtrees. In all cases the attempted proofs using morasses fail for the same reason: the partial orders involved do not have strong enough closure properties.In an attempt to solve this problem Shelah and Stanley strengthened their characterization of morasses to allow applications to what they called “good canonical limit” partial orders. However, for rather subtle reasons even this strengthened forcing axiom is not good enough for the proposed applications. The problem this time is that Shelah and Stanley's “weak commutativity of Lim and restriction” requirement (see [9, 3.9(iv)]) is not satisfied. Furthermore, there is reason to believe that an ordinary morass is just not good enough for these applications, since in L morasses exist at all regular uncountable cardinals, but even a weak form of Todorčević's conclusion cannot hold at ineffable cardinals (see the end of §4).A possible solution to this problem is suggested by the fact that □κ is equivalent to a forcing axiom which applies to partial orders satisfying precisely the kind of weak closure conditions involved in the examples described above (see [13]). What is needed to make the proposed morass applications work is something which will do for morass constructions what □κ does for ordinary transfinite recursion constructions. In this paper we show how extra structure can be built into a morass to accomplish this goal.