Iterating the cofinality-$\omega$ constructible model

TL;DR

This paper explores the properties of the cofinality-$oldsymbol{\omega}$ constructible model $C^{*}$, demonstrating its iterated behavior, equiconsistency with ZFC, and the conditions for infinite decreasing sequences involving large cardinals.

Contribution

It establishes the equiconsistency of $C^{*}$ with ZFC and analyzes the behavior of iterated $C^{*}$ models, including the necessity of measurable cardinals for infinite sequences.

Findings

$C^{*}$ is equiconsistent with ZFC.

Finite decreasing sequences of iterated $C^{*}$s exist.

Infinite decreasing sequences require a measurable cardinal.

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 ZFC, as well as having finite strictly decreasing sequences of iterated $C^{*}$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.