We investigate the relation between C ⁎ , the model of sets constructible using first order logic augmented with the “cofinality- ω ” quantifier, and “short” sequences of measures – sequences of measures of order 1, which are shorter than their minimum. We show that certain core models for short sequences of measures are contained in C ⁎ ; we compute C ⁎ in a model of the form L [ U ] where U is a short sequence of measures, and in models of the form L [ U ] [ G ] where G is generic for adding Prikry sequences to some of the measurables of U ; and prove that if there is an inner model with a short sequence of measures of order type χ , then there is such an inner model in C ⁎ .

Abstract We 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.

Abstract We consider the modality “ $\varphi $ is true in every $\sigma $ -centered forcing extension,” denoted $\square \varphi $ , and its dual “ $\varphi $ is true in some $\sigma $ -centered forcing extension,” denoted $\lozenge \varphi $ (where $\varphi $ is a statement in set theory), which give rise to the notion of a principle of $\sigma $ -centered forcing. We prove that if ZFC is consistent, then the modal logic of $\sigma $ -centered forcing, i.e., the ZFC-provable principles of $\sigma $ -centered forcing, is exactly $\mathsf {S4.2}$ . We also generalize this result to other related classes of forcing.