The goal is to find a model in which CH is false and which is canonical in the sense that its theory cannot be altered by set forcing in the presence of large cardinals. The background motivation is this: First, we know that in the presence of large cardinal axioms the theory of second-order arithmetic and even the entire theory of L(ℝ) is invariant under set forcing. The importance of this is that it demonstrates that our main independence techniques cannot be used to establish the independence of questions about second-order arithmetic (or about L(ℝ)) in the presence of large cardinals. Second, experience has shown that the large cardinal axioms in question seem to answer all of the major known open problems about second-order arithmetic and L(ℝ) and the set forcing invariance theorems give precise content to the claim that these axioms are “effectively complete”.[5] 