Title: The downward directed grounds hypothesis and very large cardinals,
Abstract: 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.