We investigate the relation between $C^{ *}$, the model of sets constructible using first order logic augmented with the "cofinality-$\omega$" 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 $L\left[\mathcal{ U}\right]$ where $\mathcal{ U}$ is a short sequence of measures, and in models of the form $L\left[\mathcal{ U}\right]\left[G\right]$ where $G$ is generic for adding Prikry sequences to some of the measurables of $\mathcal{ U}$; and prove that if there is an inner model with a short sequence of measures of order type $\chi$, then there is such an inner model in $C^{ *}$.
Categories: math.LO