ITERATING THE COFINALITY- CONSTRUCTIBLE MODEL

Abstract

AbstractWe investigate iterating the construction of $C^{*}$ , the L-like inner model constructed using first order logic augmented with the “cofinality $\omega $ ” quantifier. We first show that $\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$ is equiconsistent with $\mathrm {ZFC}$ , 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 $\omega $ , and that an inner model with a measurable cardinal is required for that.