HOD denotes the class of *Hereditarily Ordinal Definable* sets. It is a definable canonical inner [model](Model.md "Model") of [ZFC](ZFC.md "ZFC"). Although it is definable, this definition is not absolute for transitive inner models of ZF, i.e. given two models $M$ and $N$ of $ZF$, $HOD$ as computed in $M$ may differ from $HOD$ as computed in $N$. ## Ordinal Definable Sets Elements of $OD$ are all definable from a finite collection of ordinals. ## Relativizations ## gHOD Generic HOD (gHOD) is the intersection of HODs of all set-[generic](Forcing.md "Forcing") extensions of $V$. {% cite Fuchs2015 %} This article is a stub. Please help us to improve Cantor's Attic by adding information.