| Assuming the axiom of determinacy, we give a new proof of the strong partition relation on w1 . Further, we present a streamlined proof that J <lambda+(a) (the ideal of sets which force cof product a ≤ lambda) is generated from J<lambda (a) by adding a singleton. Combining these results with a polarized partition relation on ℵw1 and a covering result of Woodin, we obtain our main result that if the nonstationary ideal on w1 is w2 saturated and there are at least w many Woodin cardinals plus a measurable cardinal above them all, then some regular cardinal <ℵw2 in L[ R ] must collapse in V. |
