Recall that one key virtue of large cardinal axioms is that they “effectively settle” the theory of second-order arithmetic (and, in fact, the theory of L(ℝ) and more) in the sense that in the presence of large cardinals one cannot use the method of set forcing to establish independence with respect to statements about L(ℝ). This notion of invariance under set forcing played a key role in Section 3.1. We can now rephrase this notion in terms of Ω-logic. 