In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using recursive ordinal notations.
The smallest non-recursive ordinal is the ChurchâÂÂKleene ordinal, , named after Alonzo Church and S. C. Kleene; its order type is the set of all recursive ordinals. Since the successor of a recursive ordinal is recursive, the ChurchâÂÂKleene ordinal is a limit ordinal. It is also the smallest ordinal that is not hyperarithmetical, and the smallest admissible ordinal after (an ordinal is called admissible if .) The -recursive subsets of are exactly the subsets of .
The notation is in reference to , the first uncountable ordinal, which is the set of all countable ordinals, analogously to how the ChurchâÂÂKleene ordinal is the set of all recursive ordinals. Some old sources use to denote the ChurchâÂÂKleene ordinal.
For a set , a set is -computable if it is computable from a Turing machine with an oracle that queries . The relativized ChurchâÂÂKleene ordinal is the supremum of the order types of -computable relations. The FriedmanâÂÂJensenâÂÂSacks theorem states that for every countable admissible ordinal , there exists a set such that .
, first defined by Stephen G. Simpson is an extension of the ChurchâÂÂKleene ordinal. This is the smallest limit of admissible ordinals, yet this ordinal is not admissible. Alternatively, this is the smallest such that is a model of -comprehension.
The th admissible ordinal is sometimes denoted by .
Recursively [X] ordinals, where [X] typically represents a large cardinal property, are kinds of nonrecursive ordinals. Rathjen has called these ordinals the "recursively large counterparts" of X, however the use of "recursively large" here is not to be confused with the notion of an ordinal being recursive.
An ordinal is called recursively inaccessible if it is admissible and a limit of admissibles. Alternatively, is recursively inaccessible iff is the th admissible ordinal, or iff , an extension of KripkeâÂÂPlatek set theory stating that each set is contained in a model of KripkeâÂÂPlatek set theory. Under the condition that ("every set is hereditarily countable"), is recursively inaccessible iff is a model of -comprehension.
An ordinal is called recursively hyperinaccessible if it is recursively inaccessible and a limit of recursively inaccessibles, or where is the th recursively inaccessible. Like "hyper-inaccessible cardinal", different authors conflict on this terminology.
An ordinal is called recursively Mahlo if it is admissible and for any -recursive function there is an admissible such that (that is, is closed under ). Mirroring the Mahloness hierarchy, is recursively -Mahlo for an ordinal if it is admissible and for any -recursive function there is an admissible ordinal such that is closed under , and is recursively -Mahlo for all .
An ordinal is called recursively weakly compact if it is -reflecting, or equivalently, 2-admissible. These ordinals have strong recursive Mahloness properties, if ñ is -reflecting then is recursively -Mahlo.
An ordinal is stable if is a -elementary-substructure of , denoted . These are some of the largest named nonrecursive ordinals appearing in a model-theoretic context, for instance greater than for any computably axiomatizable theory .<sup>Proposition 0.7</sup>. There are various weakenings of stable ordinals:
Even larger nonrecursive ordinals include: