The downward directed grounds hypothesis and very large cardinals

摘要

A transitive model M of ZFC is called a ground if the universe V is a set, forcing extension of M. We show that the grounds of V are downward set-directed. (Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) V has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must, be a ground.