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HOD denotes the class of Hereditarily Ordinal Definable sets. It is a definable canonical inner model of 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.



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