Iterating the cofinality-$\omega$ constructible model

Abstract

We investigate iterating the construction of $C^{\ast }$, the $L$-like inner model constructed using first order logic augmented with the "cofinality $\omega$" quantifier. We first show that ${\left(C^{\ast }\right)}^{C^{\ast }}=C^{\ast }\ne L$ is equiconsistent with ZFC, as well as having finite strictly decreasing sequences of iterated $C^{\ast }$s. We then show that in models of the form $L^{\mu }$ we get infinite decreasing sequences of length $\omega$, and that an inner model with a measurable cardinal is required for that.