Iterated Class Forcing

In this paper we develop the notion of \stratiied" class forcing and show that this property both implies coonality-preservation and is preserved by iterations with the appropriate support. Many Easton-style and Jensen-style forcings are stratiied, as are some more exotic forcings obtained by mixing these types together (see Easton 70], section 36 of Jech 78], Beller-Jensen-Welch 82], Friedman 90]). As a sample application, coonalities are preserved by an iteration of length ORD where at even stages i, an Easton-style forcing adds a Cohen set to regular cardinals card (i), at odd stages i + 1 the class added at stage i is coded by a subset of the least innnite regular cardinal card(i) via the techniques of Friedman 94A] or Friedman 94B], and for any regular , fijp(i) is nontrivial below g is a subset of + of size < , for each condition p in the iteration. Jensen coding as in Beller-Jensen-Welch 82] is not stratiied but obeys a related property, called-stratiication, which is also preserved by iterations with the appropriate (larger) support. As a sample application, the original form of Jensen coding can be used in the iteration of the preceding paragraph, provided the Cohen sets are added at successor cardinals only, full support is used and the condition stated at the end of that paragraph is imposed only at successor cardinals. We now deene stratiication, in the language of GG odel-Bernays class theory. Deenition P (partially ordered by) is stratiied if there is a class A such that V = LA], hV; Ai has a-deenable well ordering and: (a) P and are A-deenable. A condition in P is a function p on an initial segment of Card = f0gg Innnite Cardinals, where if q extends p as a function, q() = ; for all 2 Dom(q) ? Dom(p), then we identify p with q. Also we require that p() = ; for singular and the conditions with constant value ; are the weakest in P. Lastly, fpjp + 2 H + g is dense for each 2 Card. (b) (Density Reduction) Let be regular and deene p q if (p q and p