Ordinal definable set

In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first order formula. Ordinal definable sets were introduced by Gödel (1965).

A drawback to this informal definition is that requires quantification over all first order formulas, which cannot be formalized in the language of set theory. However there is a different way of stating this which can be so formalized. By this way, a set S is formally defined to be ordinal definable if there some collection of ordinals α₁...α_n such that $S \isin V_{\alpha_1}$ and can be defined as a element of $V_{\alpha_1}$ by a first-order formula φ taking α₂...α_n as parameters. Here $V_{\alpha_1}$ denotes the set indexed by the ordinal α₁ in the von Neumann hierarchy of sets. In other words, S is the unique object such that φ(S, α₂...α_n) holds with its quantifiers ranging over $V_{\alpha_1}$.

The class of all ordinal definable sets is denoted OD; it is not necessarily transitive, and need not be a model of ZFC because it might not satisfy the axiom of extensionality. A set is hereditarily ordinal definable if it is ordinal definable and all elements of its transitive closure are ordinal definable. The class of hereditarily ordinal definable sets is denoted by HOD, and is a transitive model of ZFC, with a definable well ordering. It is consistent with the axioms of set theory that all sets are ordinal definable, and so hereditarily ordinal definable. The assertion that this situation holds is referred to as V = OD or V = HOD. It follows from V = L, and implies that the universe can be well-ordered.

References

• Gödel, Kurt (1965) [1946], "Remarks before the Princeton Bicentennial Conference on Problems in Mathematics", in Davis, Martin, The undecidable. Basic papers on undecidable propositions, unsolvable problems and computable functions, Raven Press, Hewlett, N.Y., pp. 84–88, ISBN 978-0486432281, MR0189996 
• Kunen, Kenneth (1980), Set theory: An introduction to independence proofs, Elsevier, ISBN 978-0-444-86839-8 

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