HOD
From Cantor's Attic
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 $V$.[1]
References
- Fuchs, Gunter and Hamkins, Joel David and Reitz, Jonas. Set-theoretic geology. Annals of Pure and Applied Logic 166(4):464 - 501, 2015. www arχiv DOI bibtex
This article is a stub. Please help us to improve Cantor's Attic by adding information.