The enriched stable core and the relative rigidity of HOD

Sy-David Friedman¹

¹ Kurt Gödel Research Center Währinger Strasse 25 1090 Wien, Austria

Fundamenta Mathematicae, Tome 235 (2016) no. 1, pp. 1-12.

Résumé

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.

Détail

DOI : 10.4064/fm170-12-2015

Keywords: author paper v definable stable core mathbb introduced shown generic mathbb mathbb definable dense classes each v definable club contains mathbb definable club holds mathbb replaced hod where hod denotes del inner model hereditarily ordinal definable sets present article extend models class theory introducing v definable enriched stable core mathbb * * * application obtain rigidity mathbb * embeddings which constructible moreover v constructible club contains mathbb * constructible club applies model hod * therefore conclude relative v definable predicate hod rigid v constructible embeddings