Models for short sequences of measures in the cofinality-$\omega$ constructible model

TL;DR

This paper explores the structure of the constructible universe $C^{*}$ with the cofinality-$\omega$ quantifier, showing it contains core models with short measure sequences and analyzing its behavior in various models with Prikry forcing.

Contribution

It demonstrates that core models with short measure sequences are contained in $C^{*}$ and computes $C^{*}$ in models with such sequences and Prikry generic extensions.

Findings

Core models with short measure sequences are in $C^{*}$.

$C^{*}$ can be computed in models with short measure sequences.

Inner models with short measure sequences exist within $C^{*}$.

Abstract

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 $C^{*}$.