HOD

• Ordinal Definable Sets
• Relativizations
• gHOD

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.

Relativizations

gHOD

Generic HOD (gHOD) is the intersection of HODs of all set-generic extensions of $V$. (Fuchs et al., 2015)

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References

1. Fuchs, G., Hamkins, J. D., & Reitz, J. (2015). Set-theoretic geology. Annals of Pure and Applied Logic, 166(4), 464–501. https://doi.org/http://web.archive.org/web/20191116153209/https://doi.org/10.1016/j.apal.2014.11.004

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