Structural Properties of the Stable Core

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] .