mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

, a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''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 . A drawback to this informal definition is that it 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 the definition that can be so formalized. In this approach, a set ''S'' is formally defined to be ordinal definable if there is some collection of ordinals ''α''₁, ..., ''α''_''n'' such that

S \isin V_

and

S

can be defined as an element of

V_

by a first-order formula φ taking α₂, ..., α_''n'' as parameters. Here

V_

denotes the set indexed by the ordinal ''α''₁ in the von Neumann hierarchy. In other words, ''S'' is the unique object such that φ(''S'', α₂...α_''n'') holds with its quantifiers ranging over

V_

. The

class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...

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 In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements ...

. A set is hereditarily ordinal definable if it is ordinal definable and all elements of its

transitive closure In mathematics, the transitive closure of a binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite s ...

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 is equivalent to the existence of a (definable)

well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-orde ...

ing of the universe. Note however that the formula expressing V = HOD need not hold true within HOD, as it is not

absolute Absolute may refer to: Companies * Absolute Entertainment, a video game publisher * Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK * Absolute Software Corporation, specializes in security and data risk manage ...

for models of set theory: within HOD, the interpretation of the formula for HOD may yield an even smaller inner model. HOD has been found to be useful in that it is an

inner model In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let L = \langle \in \rangle be ...

that can accommodate essentially all known

large cardinal In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least � ...

s. This is in contrast with the situation for

core model In set theory, the core model is a definable inner model of the von Neumann universe, universe of all Set (mathematics), sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a ...

s, as core models have not yet been constructed that can accommodate

supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...

s, for example.

References

* * Set theory {{settheory-stub