The enriched stable core and the relative rigidity of HOD
Volume 235 / 2016
Abstract
In the author’s 2012 paper, the $V$-definable Stable Core ${\mathbb {S}}=(L[S],S)$ was introduced. It was shown that $V$ is generic over ${\mathbb {S}}$ (for ${\mathbb {S}}$-definable dense classes), each $V$-definable club contains an ${\mathbb {S}}$-definable club, and the same holds with ${\mathbb {S}}$ replaced by $({\rm HOD},S)$, where ${\rm HOD}$ denotes Gödel’s inner model of hereditarily ordinal-definable sets. In the present article we extend this to models of class theory by introducing the $V$-definable Enriched Stable Core ${\mathbb {S}}^*=(L[S^*],S^*)$. As an application we obtain the rigidity of ${\mathbb {S}}^*$ for all embeddings which are “constructible from $V$”. Moreover, any “$V$-constructible” club contains an “${\mathbb {S}}^*$-constructible” club. This also applies to the model $({\rm HOD},S^*)$, and therefore we conclude that, relative to a $V$-definable predicate, ${\rm HOD}$ is rigid for $V$-constructible embeddings.
