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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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244:if it is ordinal definable and all elements of its
329:Set theory: An introduction to independence proofs
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267:HOD has been found to be useful in that it is an
302:, Raven Press, Hewlett, N.Y., pp. 84–88,
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53:. Ordinal definable sets were introduced by
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78:if there is some collection of ordinals
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104:as parameters that uniquely defines
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203:{\displaystyle V_{\alpha _{1}}}
151:{\displaystyle V_{\alpha _{1}}}
242:hereditarily ordinal definable
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235:axiom of extensionality
218:indexed by the ordinal
281:supercompact cardinals
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216:von Neumann hierarchy
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240:A set further is
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117:{\displaystyle S}
76:ordinal definable
43:ordinal definable
18:Ordinal definable
16:(Redirected from
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277:core models
269:inner model
360:Set theory
287:References
231:transitive
61:Definition
32:set theory
190:α
138:α
354:Category
333:Elsevier
327:(1980),
262:absolute
98:, ..., α
47:ordinals
318:0189996
298:(ed.),
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