Structural Properties of the Stable Core

TL;DR

This paper investigates the structural properties of the stable core, an inner model related to $L[S]$, revealing its complex interactions with large cardinals and its potential to exhibit diverse set-theoretic phenomena.

Contribution

It provides new insights into the stable core's structure, including its ability to have GCH fail everywhere and contain large cardinals, expanding understanding of inner models with large cardinal features.

Findings

GCH can fail at all regular cardinals in the stable core

The stable core can have a proper class of measurable cardinals

Measurable cardinals may not be downward absolute to the stable core

Abstract

The stable core, an inner model of the form $\langle L[S],\in, S\rangle$ for a simply definable predicate $S$, was introduced by the first author in [Fri12], where he showed that $V$ is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname{GCH}$ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in $V$, then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots, C_n]$ for specially nested class clubs $C_1, \dots, C_n$, like those arising in the stable core, generalizing recent results of Welch [Wel19].