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A drawback to the above informal definition is that it requires quantification over all first-order formulas, which cannot be formalized in the standard language of set theory. However, there is a different, formal such characterization:
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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
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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.
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for models of set theory: within HOD, the interpretation of the formula for HOD may yield an even smaller inner model.
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of the universe. Note however that the formula expressing V = HOD need not hold true within HOD, as it is not
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Gödel, Kurt (1965) , "Remarks before the
Princeton Bicentennial Conference on Problems in Mathematics", in
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The undecidable. Basic papers on undecidable propositions, unsolvable problems and computable functions
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of all ordinal definable sets is denoted OD; it is not necessarily
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if, informally, it can be defined in terms of a finite number of
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233:if it is ordinal definable and all elements of its
318:Set theory: An introduction to independence proofs
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256:HOD has been found to be useful in that it is an
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42:. Ordinal definable sets were introduced by
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67:if there is some collection of ordinals
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192:{\displaystyle V_{\alpha _{1}}}
140:{\displaystyle V_{\alpha _{1}}}
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298:978-0-486-43228-1
229:A set further is
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106:{\displaystyle S}
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266:core models
258:inner model
349:Set theory
276:References
220:transitive
50:Definition
21:set theory
179:α
127:α
343:Category
322:Elsevier
316:(1980),
251:absolute
87:, ..., α
36:ordinals
307:0189996
287:(ed.),
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293:ISBN
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