#paper # Abstract We investigate iterating the construction of C ∗, the L-like inner model constructed using first order logic augmented with the “cofinality ” quantifier. We first show that (C ∗)C ∗ = C ∗ = L is equiconsistent with ZFC, as well as having finite strictly decreasing sequences of iterated C ∗s. We then show that in models of the form L[U ] we get infinite decreasing sequences of length , and that an inner model with a measurable cardinal is required for that.